Example 1.1. If \(f:{\mathbb Q}\rightarrow {\mathbb Q}\) is given by

\begin{alignat*}{2}
f(x)&=-1&&\qquad\text{if $x^{2}<2$,}\\
f(x)&=1&&\qquad\text{otherwise,}
\end{alignat*}

then

(i) \(f\) is continuous function with \(f(0)=-1\), \(f(2)=1\) yet there does not exist a \(c\) with \(f(c)=0\),

(ii) \(f\) is a differentiable function with \(f'(x)=0\) for all \(x\) yet \(f\) is not constant.

Definition 1.2. If \(a_{n}\in {\mathbb R}\) for each \(n\geq 1\) and \(a\in {\mathbb R}\) then we say that \(a_{n}\rightarrow a\) if given \(\epsilon >0\) we can find an \(n_{0}(\epsilon )\) such that \[|a_{n}-a|<\epsilon \ \text {for all $n\geq n_{0}(\epsilon )$}.\]

Axiom 1.3 (The fundamental axiom of analysis). If \(a_{n}\in {\mathbb R}\) for each \(n\geq 1\), \(A\in {\mathbb R}\) and \(a_{1}\leq a_{2}\leq a_{3}\leq \ldots \) and \(a_{n}<A\) for each \(n\) then there exists an \(a\in {\mathbb R}\) such that \(a_{n}\rightarrow a\) as \(n\rightarrow \infty \).

Lemma 2.1. (i) The limit is unique. That is, if \(a_{n}\rightarrow a\) and \(a_{n}\rightarrow b\) as \(n\rightarrow \infty \) then \(a=b\).

(ii) If \(a_{n}\rightarrow a\) as \(n\rightarrow \infty \) and \(n(1)<n(2)<n(3)\ldots \) then \(a_{n(j)}\rightarrow a\) as \(j\rightarrow \infty \).

(iii) If \(a_{n}=c\) for all \(n\) then \(a_{n}\rightarrow c\) as \(n\rightarrow \infty \).

(iv) If \(a_{n}\rightarrow a\) and \(b_{n}\rightarrow b\) as \(n\rightarrow \infty \) then \(a_{n}+b_{n}\rightarrow a+b\).

(v) If \(a_{n}\rightarrow a\) and \(b_{n}\rightarrow b\) as \(n\rightarrow \infty \) then \(a_{n}b_{n}\rightarrow ab\).

(vi) If \(a_{n}\rightarrow a\) as \(n\rightarrow \infty \) and \(a_{n}\neq 0\) for each \(n\), \(a\neq 0\) then \(a_{n}^{-1}\rightarrow a^{-1}\).

(vii) If \(a_{n}\leq A\) for each \(n\) and \(a_{n}\rightarrow a\) as \(n\rightarrow \infty \) then \(a\leq A\).

Lemma 2.2. A decreasing sequence of real numbers bounded below tends to a limit.

Theorem 2.3 (Axiom of Archimedes). \[\frac {1}{n}\rightarrow 0\ \text {as $n\rightarrow \infty $}\]

Theorem 2.4. Given any real number \(K\) we can find an integer \(n\) with \(n>K\).

Theorem 3.1 (Bolzano–Weierstrass). If \(x_{n}\in {\mathbb R}\) and there exists a \(K\) such that \(|x_{n}|\leq K\) for all \(n\) then we can find \(n(1)<n(2)<\ldots \) and \(x\in {\mathbb R}\) such that \(x_{n(j)}\rightarrow x\) as \(j\rightarrow \infty \).

Lemma 3.2. If \(x_{n}\in {\mathbb R}\) then at least one of the following two statements must be true.

(A) There exist \(n(1)<n(2)<\ldots \) such that \(x_{n(j)}\leq x_{n(j+1)}\) for each \(j\geq 1\).

(B) There exist \(m(1)<m(2)<\ldots \) such that \(x_{m(j)}\geq x_{m(j+1)}\) for each \(j\geq 1\).

Definition 3.3. Consider a non-empty set \(A\) of real numbers. We say that \(\alpha \) is a least upper bound (or supremum) for \(A\) if the following two conditions hold.

(i) \(\alpha \geq a\) for all \(a\in A\) (that is, \(\alpha \) is an upper bound).

(ii) If \(\beta \geq a\) for all \(a\in A\) then \(\beta \geq \alpha \) (that is, \(\alpha \) is no greater than any possible upper bound).

Lemma 3.4. The supremum is unique if it exists.

Lemma 3.5. Consider a non-empty set \(A\) of real numbers; \(\alpha \) is a least upper bound for \(A\) if and only if the following two conditions hold.

(i) \(\alpha \geq a\) for all \(a\in A\).

(ii) Given \(\epsilon >0\) there exists an \(a\in A\) such that \(a+\epsilon \geq \alpha \).

Theorem 3.6 (Existence of the supremum). If \(A\) is a non empty set of real numbers which is bounded above (that is, there exists a \(K\) such that \(a\leq K\) for all \(a\in A\)) then \(A\) has a supremum.

Lemma 3.7. We work in \(\mathbb C\). If \(\sum _{n=0}^{\infty }a_{n}z^{n}\) converges for \(z=z_{0}\) then it converges for all \(z\) with \(|z|<|z_{0}|\).

Theorem 3.8. We work in \(\mathbb C\). Consider the sum \(\sum _{n=0}^{\infty }a_{n}z^{n}\). Either \(\sum _{n=0}^{\infty }a_{n}z^{n}\) converges for all \(z\) (in this case we say that the series has infinite radius of convergence) or there exists a real number \(R\geq 0\) such that

(i) \(\sum _{n=0}^{\infty }a_{n}z^{n}\) converges for all \(|z|<R\),

(ii) \(\sum _{n=0}^{\infty }a_{n}z^{n}\) diverges for all \(|z|>R\),

(in this case we say that the series has radius of convergence R).

Example 3.9. (i) The sum \(\sum _{n=1}^{\infty }n^{-2}z^{n}\) has radius of convergence \(1\) and converges for all \(|z|=1\).

(ii) The sum \(\sum _{n=0}^{\infty }z^{n}\) has radius of convergence \(1\) and diverges for all \(|z|=1\).

(iii) The sum \(\sum _{n=1}^{\infty }n^{-1}z^{n}\) has radius of convergence \(1\), diverges for \(z=1\) and converges for \(z=-1\).

Definition 3.10. We say that a sequence of real numbers \(x_{n}\) is a Cauchy sequence if given any \(\epsilon >0\) we can find \(n_{0}(\epsilon )\) such that \(|x_{p}-x_{q}|<\epsilon \) whenever \(p,q\geq n_{0}(\epsilon )\).

Lemma 3.11. Any convergent sequence forms a Cauchy sequence.

Theorem 3.12. Any Cauchy sequence of real numbers converges.

Theorem 3.13 (General principle of convergence). A sequence of real numbers converges if and only if it is a Cauchy sequence.

Exercise 3.14 (Uncountability of the reals). Let \(y_{1}\), \(y_{2}\), …be any sequence of points in \(\mathbb R\). Let \(x_{0}=0\), \(\delta _{0}=1\).

(i) Show that you can construct inductively a sequence of real numbers \(x_{1}\), \(x_{2}\), …and positive numbers \(\delta _{j}\) such that

  (a) \(|x_{n}-x_{n-1}|<\delta _{n-1}/4\),

  (b) \(x_{n}\neq y_{n}\),

  (c) \(0<\delta _{n}<|x_{n}-y_{n}|\),

  (d) \(\delta _{n}<\delta _{n-1}/4\).

(ii) Show that \(\delta _{n+m}<4^{-m}\delta _{n}\) for \(m,n\geq 0\) and deduce that the \(x_{n}\) form a Cauchy sequence. Conclude that \(x_{n}\) tends to a limit \(x\).

(iii) Show that \(|x_{n+m}-x_{n}|<\delta _{n}/3\) for all \(m,n\geq 0\). Deduce that \(|x-x_{n}|\leq \delta _{n}/3\) for all \(n\geq 0\). Why does this show that \(y_{n}\neq x\) for all \(n\)?

(iv) Deduce that the real numbers are uncountable.

Lemma 4.1. If \(\|\ \|\) is the Euclidean norm on \({\mathbb R}^{m}\) then

(i) \(\|{\mathbf x}\|\geq 0\) for all \({\mathbf x}\in {\mathbb R}^{m}\),

(ii) If \(\|{\mathbf x}\|= 0\) then \({\mathbf x}={\mathbf 0}\),

(iii) If \(\lambda \in {\mathbb R}\) and \({\mathbf x}\in {\mathbb R}^{m}\) then \(\|\lambda {\mathbf x}\|=|\lambda |\|{\mathbf x}\|\).

(iv) (The triangle inequality) If \({\mathbf x},{\mathbf y}\in {\mathbb R}^{n}\) then \(\|{\mathbf x}+{\mathbf y}\| \leq \|{\mathbf x}\|+\|{\mathbf y}\|\).

Definition 4.2. If \(\mathbf {a}_{n}\in {\mathbb R}^{m}\) for each \(n\geq 1\) and \(\mathbf {a}\in {\mathbb R}^{m}\) then we say that \(\mathbf {a}_{n}\rightarrow \mathbf {a}\) if given \(\epsilon >0\) we can find an \(n_{0}(\epsilon )\) such that \[\|\mathbf {a}_{n}-\mathbf {a}\|<\epsilon \ \text {for all $n\geq n_{0}(\epsilon )$}.\]

Lemma 4.3. (i) The limit is unique. That is, if \(\mathbf {a}_{n}\rightarrow \mathbf {a}\) and \(\mathbf {a}_{n}\rightarrow \mathbf {b}\) as \(n\rightarrow \infty \) then \(\mathbf {a}=\mathbf {b}\).

(ii) If \(\mathbf {a}_{n}\rightarrow \mathbf {a}\) as \(n\rightarrow \infty \) and \(n(1)<n(2)<n(3)\ldots \) then \(\mathbf {a}_{n(j)}\rightarrow \mathbf {a}\) as \(j\rightarrow \infty \).

(iii) If \(\mathbf {a}_{n}=\mathbf {c}\) for all \(n\) then \(\mathbf {a}_{n}\rightarrow \mathbf {c}\) as \(n\rightarrow \infty \).

(iv) If \(\mathbf {a}_{n}\rightarrow \mathbf {a}\) and \(\mathbf {b}_{n}\rightarrow \mathbf {b}\) as \(n\rightarrow \infty \) then \(\mathbf {a}_{n}+\mathbf {b}_{n} \rightarrow \mathbf {a}+\mathbf {b}\).

(v) Suppose \(\mathbf {a}_{n}\in {\mathbb R}^{m}\), \(\mathbf {a}\in {\mathbb R}^{m}\), \(\lambda _{n}\in {\mathbb R}\) and \(\lambda \in {\mathbb R}\). If \(\mathbf {a}_{n}\rightarrow \mathbf {a}\) and \(\lambda _{n}\rightarrow \lambda \) then \(\lambda _{n}\mathbf {a}_{n}\rightarrow \lambda \mathbf {a}\).

Theorem 4.4 (Bolzano–Weierstrass). If \(\mathbf {x}_{n}\in {\mathbb R}^{m}\) and there exists a \(K\) such that \(\|\mathbf {x}_{n}\|\leq K\) for all \(n\) then we can find \(n(1)<n(2)<\ldots \) and \(\mathbf {x}\in {\mathbb R}\) such that \(\mathbf {x}_{n(j)}\rightarrow \mathbf {x}\) as \(j\rightarrow \infty \).

Definition 4.5. We say that a sequence of points \(\mathbf {x}_{n}\) in \({\mathbb R}^{m}\) is a Cauchy sequence if given any \(\epsilon >0\) we can find \(n_{0}(\epsilon )\) such that \(\|\mathbf {x}_{p}-\mathbf {x}_{q}\|<\epsilon \) for all \(p,q\geq n_{0}(\epsilon )\).

Theorem 4.6 (General Principle of Convergence). A sequence in \({\mathbb R}^{m}\) converges if and only if it is a Cauchy sequence.

Theorem 4.7. Let \(\mathbf {a}_{n}\in {\mathbb R}^{m}\) for each \(n\). If \(\sum _{j=1}^{\infty }\|\mathbf {a}_{j}\|\) converges then so does \(\sum _{j=1}^{\infty }\mathbf {a}_{j}\).

Definition 5.1. A set \(F\subseteq {\mathbb R}^{m}\) is closed if whenever \(\mathbf {x}_{n}\in F\) for each \(n\) and \(\mathbf {x}_{n}\rightarrow \mathbf {x}\) as \(n\rightarrow \infty \) then \(\mathbf {x}\in F\).

Theorem 5.2. If \(K\) is closed bounded set in \({\mathbb R}^{m}\) then every sequence in \(K\) has a subsequence converging to a point of \(K\).

Definition 5.3. A set \(U\subseteq {\mathbb R}^{m}\) is open if whenever \(\mathbf {x}\in U\) there exists an \(\epsilon ({\mathbf x})>0\) such that whenever \(\|\mathbf {x}-\mathbf {y}\|<\epsilon ({\mathbf x})\) we have \(\mathbf {y}\in U\).

Example 5.4. Consider sets in \(\mathbb R\). The interval \([a,b]=\{x\in {\mathbb R}:a\leq x \leq b\}\) is closed, the interval \((a,b)=\{x\in {\mathbb R}:a< x <b\}\) is open, the interval \([a,b)=\{x\in {\mathbb R}:a\leq x <b\}\) is neither open nor closed, \(\mathbb R\) is both open and closed.

Example 5.5. Consider sets in \({\mathbb R}^{m}\). Let \({\mathbf x}\in {\mathbb R}^{m}\) and \(r>0\).

(i) The set \(B({\mathbf x},r) =\{\mathbf {y}\in {\mathbb R}^{m}: \|\mathbf {x}-\mathbf {y}\|<r\}\) is open.

(ii) The set \(\bar {B}({\mathbf x},r) =\{\mathbf {y}\in {\mathbb R}^{m}: \|\mathbf {x}-\mathbf {y}\|\leq r\}\) is closed.

Lemma 5.6. A subset \(A\) of \({\mathbb R}^{m}\) is open if and only if each point of \(A\) is the centre of an open ball lying entirely within \(A\).

Definition 5.7. The set \(N\) is a neighbourhood of the point \(\mathbf x\) if we can find an \(r({\mathbf x})>0\) such that \(B({\mathbf x},r({\mathbf x}))\subseteq N\).

Lemma 5.8. A subset \(A\) of \({\mathbb R}^{m}\) is open if and only if its complement \({\mathbb R}^{m}\setminus A\) is closed.

Lemma 5.9. Consider the collection \(\tau \) of open sets in \({\mathbb R}^{m}\).

(i) \(\emptyset \in \tau \), \({\mathbb R}^{m}\in \tau \).

(ii) If \(U_{\alpha }\in \tau \) for all \(\alpha \in A\) then \(\bigcup _{\alpha \in A} U_{\alpha }\in \tau \).

(iii) If \(U_{1},U_{2},\dots ,U_{n}\in \tau \) then \(\bigcap _{j=1}^{n}U_{j}\in \tau \).

Lemma 5.10. Consider the collection \(\mathcal {F}\) of closed sets in \({\mathbb R}^{m}\).

(i) \(\emptyset \in \mathcal {F}\), \({\mathbb R}^{m}\in \mathcal {F}\).

(ii) If \(F_{\alpha }\in \mathcal {F}\) for all \(\alpha \in A\) then \(\bigcap _{\alpha \in A} F_{\alpha }\in \mathcal {F}\).

(iii) If \(F_{1},F_{2},\dots ,F_{n}\in \mathcal {F}\) then \(\bigcup _{j=1}^{n}F_{j}\in \mathcal {F}\).

Example 5.11. (i) \(\bigcap _{j=1}^{\infty }(-2-j^{-1},2+j^{-1}) =[-2,2]\), \(\bigcap _{j=1}^{\infty }(-2+j^{-1},2-j^{-1}) =(-1,1)\), \(\bigcap _{j=1}^{\infty }(-2+j^{-1},2+j^{-1})=(-1,2]\).

(ii) \(\bigcup _{j=1}^{\infty }[-1-j^{-1},1+j^{-1}] =[-2,2]\), \(\bigcup _{j=1}^{\infty }[-1+j^{-1},1-j^{-1}] =(-1,1)\), \(\bigcup _{j=1}^{\infty }[-1+j^{-1},1+j^{-1}]=(-1,2]\).

Definition 5.12. Let \(E\subseteq {\mathbb R}^{m}\) we say that a function \(f:E\rightarrow {\mathbb R}^{p}\) is continuous at some point \({\mathbf x}\in E\) if given \(\epsilon >0\) we can find a \(\delta (\epsilon ,{\mathbf x})>0\) such that whenever \({\mathbf y}\in E\) and \(\|{\mathbf x}-{\mathbf y}\|<\delta (\epsilon ,{\mathbf x})\) we have \[\|f({\mathbf x})-f({\mathbf y})\|<\epsilon .\] If \(f\) is continuous at every point \({\mathbf x}\in E\) we say that \(f\) is a continuous function on \(E\).

Exercise 5.13. After looking at parts (iii) to (v) of Lemma 4.3 state the corresponding results for continuous functions. (Thus part (v) corresponds to the statement that if \(\lambda :E\rightarrow {\mathbb R}\) and \(f:E\rightarrow {\mathbb R}^{p}\) are continuous at \({\mathbf x}\in E\) then so is \(\lambda f\).) Prove your statements directly from Definition 5.12. Consult your supervisor if you have any problems.

Lemma 5.14. Let \(E\subseteq {\mathbb R}^{m}\) and suppose that the function \(f:E\rightarrow {\mathbb R}^{p}\) is continuous at \({\mathbf x}\in E\). Then if \({\mathbf x}_{n}\in E\) and \({\mathbf x}_{n}\rightarrow {\mathbf x}\) it follows that \(f({\mathbf x}_{n})\rightarrow f({\mathbf x})\).

Lemma 5.15. The function \(f:{\mathbb R}^{m}\rightarrow {\mathbb R}^{p}\) is continuous if and only if \(f^{-1}(O)\) is open whenever \(O\) is open.

Lemma 5.16. If \(f:{\mathbb R}^{m}\rightarrow {\mathbb R}^{p}\) and \(g:{\mathbb R}^{p}\rightarrow {\mathbb R}^{g}\) are continuous the so is their composition \(g\circ f\).

Theorem 5.17. Suppose that \(F_{1}\), \(F_{2}\), …are non-empty bounded closed sets in \({\mathbb R}^{m}\) such that \(F_{1}\supseteq F_{2}\supseteq \dots \). Then \(\bigcap _{j=1}^{\infty }F_{j}\neq \emptyset .\)

Example 5.18. The set \(\{(x,0)\in {\mathbb R}^{2}:0<x<1\}\) is not open in \({\mathbb R}^{2}\).

Theorem 6.1 (Intermediate value theorem). If \(f:[a,b]\rightarrow {\mathbb R}\) is continuous and \(f(a)\leq 0\leq f(b)\) then there exists a \(c\in [a,b]\) such that \(f(c)=0\).

Example 6.2. If \(f:[0,1]\rightarrow {\mathbb C}\) is given by \(f(t)=\exp \pi it\) then \(f\) is continuous, \(f(0)=1\), \(f(1)=-1\) but \(f(t)\neq 0\) for all \(t\).

Theorem 6.3. Let \(K\) be a closed bounded subset of \({\mathbb R}^{m}\) and \(f:K\rightarrow {\mathbb R}^{p}\) a continuous function. Then \(f(K)\) is closed and bounded.

Theorem 6.4. Let \(K\) be a non-empty closed bounded subset of \({\mathbb R}^{m}\) and \(f:K\rightarrow {\mathbb R}\) a continuous function. Then we can find \({\mathbf k}_{1}\) and \({\mathbf k}_{2}\) in \(K\) such that \[f({\mathbf k}_{1})\leq f({\mathbf k})\leq f({\mathbf k}_{2})\] for all \({\mathbf k}\in K\).

Theorem 6.5. Let \(f:[a,b]\rightarrow {\mathbb R}\) be a continuous function. Then we can find \(k_{1},k_{2}\in [a,b]\) such that \[f(k_{1})\leq f(k)\leq f(k_{2})\] for all \(k\in [a,b]\)

Theorem 6.6 (Rolle’s theorem). If \(f:[a,b]\rightarrow {\mathbb R}\) is a continuous function with \(f\) differentiable on \((a,b)\) and \(f(a)=f(b)\) then we can find a \(c\in (a,b)\) such that \(f'(c)=0\).

Theorem 6.7 (The one dimensional mean value theorem).   If \(f:[a,b]\rightarrow {\mathbb R}\) is a continuous function with \(f\) differentiable on \((a,b)\) then we can find a \(c\in (a,b)\) such that \[f(b)-f(a)=(b-a)f'(c).\]

Theorem 6.8. Suppose that \(f:(\alpha ,\beta )\rightarrow {\mathbb R}\) is a differentiable function.

(i) If \(f'(x)\geq 0\) for all \(x\in (\alpha ,\beta )\) then \(f\) is an increasing function on \((\alpha ,\beta )\) (that is \(f(x)\leq f(y)\) whenever \(\alpha <x<y<\beta \)).

(ii) If \(f'(x)> 0\) for all \(x\in (\alpha ,\beta )\) then \(f\) is a strictly increasing function on \((\alpha ,\beta )\) (that is \(f(x)<f(y)\) whenever \(\alpha <x<y<\beta \)).

(iii) If \(f'(x)=0\) for all \(x\in (\alpha ,\beta )\) then \(f\) is constant.

(iv) If \(|f'(x)|\leq K\) for all \(x\in (\alpha ,\beta )\) then \(|f(u)-f(v)|\leq K|u-v|\) for all \(u,v\in (\alpha ,\beta )\)

Definition 6.9. Let \(E\subseteq {\mathbb R}^{m}\) we say that a function \(f:E\rightarrow {\mathbb R}^{p}\) is continuous on \(E\) if given any point \({\mathbf x}\in E\) and any \(\epsilon >0\) we can find a \(\delta (\epsilon ,{\mathbf x})>0\) such that whenever \({\mathbf y}\in E\) and \(\|{\mathbf x}-{\mathbf y}\|<\delta (\epsilon ,{\mathbf x})\) we have \[\|f({\mathbf x})-f({\mathbf y})\|<\epsilon .\]

Definition 6.10. Let \(E\subseteq {\mathbb R}^{m}\) we say that a function \(f:E\rightarrow {\mathbb R}^{p}\) is uniformly continuous on \(E\) if given any \(\epsilon >0\) we can find a \(\delta (\epsilon )>0\) such that whenever \({\mathbf x},{\mathbf y}\in E\) and \(\|{\mathbf x}-{\mathbf y}\|<\delta (\epsilon )\) we have \[\|f({\mathbf x})-f({\mathbf y})\|<\epsilon .\]

Example 6.11. The following three functions are continuous but not uniformly continuous.

(i) \(f_{1}:{\mathbb R}\rightarrow {\mathbb R}\) given by \(f_{1}(x)=x^{2}\).

(ii) \(f_{2}:(0,1)\rightarrow {\mathbb R}\) given by \(f_{2}(x)=x^{-1}\).

(iii) \(f_{3}:(0,1)\rightarrow {\mathbb R}\) given by \(f_{2}(x)=\sin (x^{-1})\).

Theorem 6.12. Let \(K\) be a closed bounded subset of \({\mathbb R}^{m}\). If \(f:K\rightarrow {\mathbb R}^{p}\) is continuous on \(K\) then \(f\) is uniformly continuous on K.

Definition 7.1. Suppose that \(E\) is a subset of of \({\mathbb R}^{m}\) and \(\mathbf x\) a point such that there exists a \(\delta >0\) with \(B({\mathbf x},\delta )\subseteq E\). We say that \(f:E\rightarrow {\mathbb R}^{p}\), is differentiable at \(\mathbf x\) if we can find a linear map \(\alpha :{\mathbb R}^{m}\rightarrow {\mathbb R}^{p}\) such that, when \(\|{\mathbf h}\|<\delta \)

\begin{equation*}
f({\mathbf x}+{\mathbf h})=f({\mathbf x})+\alpha{\mathbf h}
+{\boldsymbol\epsilon}({\mathbf x},{\mathbf h})\|{\mathbf h}\|
\tag*{$\bigstar$}
\end{equation*}

where \(\|{\boldsymbol \epsilon }({\mathbf x},{\mathbf h})\|\rightarrow 0\) as \(\|{\mathbf h\|}\rightarrow 0\). We write \(\alpha =Df({\mathbf x})\) or \(\alpha =f'({\mathbf x})\).

If \(E\) is open and \(f\) is differentiable at each point of \(E\) we say that \(f\) is differentiable on \(E\).

Lemma 7.2. Let \(f\) be as in Definition 7.1. If we use standard coordinates then if \(f\) is differentiable at \(\mathbf x\) its partial derivatives \(\displaystyle f_{i,j}({\mathbf x})= \frac {\partial f_{i}}{\partial x_{j}}\) exist and the matrix of the derivative \(Df({\mathbf x})\) is the Jacobian matrix \((f_{i,j}({\mathbf x}))\) of partial derivatives.

Lemma 7.3. If \(\alpha :{\mathbb R}^{m}\rightarrow {\mathbb R}^{p}\) is a linear map we can find a constant \(K\) such that \(|\alpha {\mathbf x}|\leq K\|{\mathbf x}\|\) for all \({\mathbf x}\in {\mathbb R}^{m}\).

Definition 7.4. If \(\alpha :{\mathbb R}^{m}\rightarrow {\mathbb R}^{p}\) is a linear map then \[\|\alpha \|=\sup _{\|{\mathbf x}\|\leq 1}\|\alpha {\mathbf x}\|.\]

Exercise 7.5. Show that if \(\alpha :{\mathbb R}^{m}\rightarrow {\mathbb R}^{p}\) is a linear map then \[\|\alpha \|=\sup _{\|{\mathbf x}\|= 1}\|\alpha {\mathbf x}\|.\]

Lemma 7.6. Let \(\alpha ,\beta :{\mathbb R}^{m}\rightarrow {\mathbb R}^{p}\) be linear maps.

(i) If \({\mathbf x}\in {\mathbb R}^{m}\) then \(\|\alpha {\mathbf x}\|\leq \|\alpha \|\,\|{\mathbf x}\|\).

(ii) \(\|\alpha \|\geq 0\),

(iii) If \(\|\alpha \|= 0\) then \(\alpha =0\),

(iv) If \(\lambda \in {\mathbb R}\) then \(\|\lambda \alpha \|=|\lambda |\|\alpha \|\).

(v) (The triangle inequality) \(\|\alpha +\beta \|\leq \|\alpha \|+\|\beta \|\).

(vi) If \(\gamma :{\mathbb R}^{p}\rightarrow {\mathbb R}^{q}\) then \(\|\gamma \alpha \|\leq \|\gamma \|\,\|\alpha \|\).

Lemma 7.7 (The chain rule). Let \(U\) be a neighbourhood of \(\mathbf x\) in \({\mathbb R}^{m}\), and \(V\) a neighbourhood of \(\mathbf x\) in \({\mathbb R}^{p}\). Suppose that \(f:U\rightarrow V\) is differentiable at \(\mathbf x\) with derivative \(\alpha \), that \(f:V\rightarrow {\mathbb R}^{q}\) is differentiable at \(\mathbf y\) with derivative \(\beta \) and that \(f({\mathbf x})={\mathbf y}\). Then \(g\circ f\) is differentiable at \(\mathbf x\) with derivative \(\beta \alpha \).

Lemma 7.8. Let \(U\) be a neighbourhood of \(\mathbf x\) in \({\mathbb R}^{m}\). Suppose that \(f,g:U\rightarrow {\mathbb R}^{p}\) are differentiable at \(\mathbf x\). Then \(f+g\) is differentiable at \(\mathbf x\) and \(D(f+g)({\mathbf x})=Df({\mathbf x})+Dg({\mathbf x})\).

Theorem 7.9 (The mean value inequality). Suppose that \(U\) is an open set in \({\mathbb R}^{m}\) and that \(f:U\rightarrow {\mathbb R}^{p}\) is differentiable. Consider the straight line segment \[L=\{(1-t){\mathbf a}+t{\mathbf b}:0\leq t\leq 1\}\] joining \(\mathbf a\) and \(\mathbf b\). If \(L\subseteq U\) (i.e. \(L\) lies entirely within \(U\)) and \(\|Df({\mathbf x})\|\leq K\) for all \({\mathbf x}\in L\) then \[\|f({\mathbf a})-f({\mathbf b})\| \leq K\|{\mathbf a}-{\mathbf b}\|.\]

Example 7.10. Let \(f:{\mathbb R}\rightarrow {\mathbb R}^{2}\) be given by \(f(t)=(\cos t,\sin t)^{T}\). Then \(f(0)=f(2\pi )\) but \(Df(t)\neq 0\) for all \(t\).

Example 7.11. Let \[U=\{{\mathbf x}\in {\mathbb R}^{2}:\|{\mathbf x}\|>1\} \setminus \{(x,0)^{T}:x\leq 0\}\}\] If we take \(\theta ({\mathbf x})\) to be the unique solution of \[\cos (\theta ({\mathbf x}))=x,\ \sin (\theta ({\mathbf x}))=y, \ -\pi <\theta (\mathbf x)<\pi \] for \({\mathbf x}=(x,y)^{T}\in U\) then \(\theta :U\rightarrow {\mathbb R}\) is everywhere differentiable with \(\|D\theta ({\mathbf x})\|<1\). However if \(\mathbf {a}=(-1,10^{-1})^{T}\), \(\mathbf {b}=(-1,-10^{-1})^{T}\) then \[|\theta (\mathbf {a})-\theta (\mathbf {b})|> \|\mathbf {a}-\mathbf {b}\|.\]

Theorem 8.1 (Continuity of partial derivatives implies differentiability). Suppose \(\delta >0\), \({\mathbf x}=(x,y)\in {\mathbb R}^{2}\), \(B({\mathbf x},\delta )\subseteq E\subseteq {\mathbb R}^{2}\) and that \(f:E\rightarrow {\mathbb R}\). If the partial derivatives \(f_{,1}\) and \(f_{,2}\) exist in \(B({\mathbf x},\delta )\) and are continuous at \(\mathbf x\) then writing \[f((x+h,y+k))=f(x,y)+f_{,1}(x,y)h+f_{,2}(x,y)k+ \epsilon (h,k)(h^{2}+k^{2})^{1/2}\] we have \(\epsilon (h,k)\rightarrow 0\) as \((h^{2}+k^{2})^{1/2}\rightarrow 0\). (In other words \(f\) is differentiable at \(\mathbf x\).)

Theorem 8.2. (i) Suppose \(\delta >0\), \({\mathbf x}\in {\mathbb R}^{m}\) \(B({\mathbf x},\delta )\subseteq E\subseteq {\mathbb R}^{m}\) and that \(f:E\rightarrow {\mathbb R}\). If the partial derivatives \(f_{,1}\), \(f_{,2}\), …\(f_{,m}\) exist in \(B({\mathbf x},\delta )\) and are continuous at \(\mathbf x\) then \(f\) is differentiable at \(\mathbf x\).

(ii) Suppose \(\delta >0\), \({\mathbf x}\in {\mathbb R}^{m}\) \(B({\mathbf x},\delta )\subseteq E\subseteq {\mathbb R}^{m}\) and that \(f:E\rightarrow {\mathbb R}^{p}\). If the partial derivatives \(f_{i,j}\) exist in \(B({\mathbf x},\delta )\) and are continuous at \(\mathbf x\) \([1\leq i\leq p,\ 1\leq j\leq m]\) then \(f\) is differentiable at \(\mathbf x\).

Theorem 8.3 (Second order Taylor series). Suppose \(\delta >0\), \({\mathbf x}=(x,y)\in {\mathbb R}^{2}\) \(B({\mathbf x},\delta )\subseteq E\subseteq {\mathbb R}^{2}\) and that \(f:E\rightarrow {\mathbb R}\). If the partial derivatives \(f_{,1}\), \(f_{,2}\), \(f_{,11}\), \(f_{,12}\), \(f_{,22}\) exist in \(B({\mathbf x},\delta )\) and are continuous at \(\mathbf x\) then writing \begin {align*} f((x+h,y+k))=&f(x,y)+f_{,1}(x,y)h+f_{,2}(x,y)k\\ &+\tfrac {1}{2}(f_{,11}(x,y)h^{2}+2f_{,12}(x,y)hk +f_{,22}(x,y)k^{2})+\epsilon (h,k)(h^{2}+k^{2}) \end {align*}

we have \(\epsilon (h,k)\rightarrow 0\) as \((h^{2}+k^{2})^{1/2}\rightarrow 0\).

Theorem 8.4 (Symmetry of second partial derivative). Suppose \(\delta >0\), \({\mathbf x}=(x,y)\in {\mathbb R}^{2}\) \(B({\mathbf x},\delta )\subseteq E\subseteq {\mathbb R}^{2}\) and that \(f:E\rightarrow {\mathbb R}\). If the partial derivatives \(f_{,1}\), \(f_{,2}\), \(f_{,11}\), \(f_{,12}\), \(f_{,21}\) \(f_{,22}\) exist in \(B({\mathbf x},\delta )\) and are continuous at \(\mathbf x\) then \(f_{,12}({\mathbf x})=f_{,21}({\mathbf x})\).

Example 8.5. If \begin {align*} f(x,y)&=\frac {xy}{(x^{2}+y^{2})^{1/2}}\qquad \text {for $(x,y)\neq (0,0)$},\\ f(0,0)&=0, \end {align*}

then \(f\) has first partial derivatives everywhere, \(f\) is differentiable except at \((0,0)\) where it is not.

Example 8.6. If \begin {align*} f(x,y)&=\frac {xy(x^{2}-y^{2})}{(x^{2}+y^{2})} \qquad \text {for $(x,y)\neq (0,0)$},\\ f(0,0)&=0, \end {align*}

then \(f\) has first and second partial derivatives everywhere but \(f_{,12}(0,0)\neq f_{,21}(0,0)\).

Theorem 8.7 (The local Taylor’s theorem). Suppose \(\delta >0\), \({\mathbf x}\in {\mathbb R}^{m}\) \(B({\mathbf x},\delta )\subseteq E\subseteq {\mathbb R}^{m}\) and that \(f:E\rightarrow {\mathbb R}^{p}\). If the all partial derivatives \(f_{i,j}\), \(f_{i,jk}\), \(f_{i,jkl}\), …exist in \(B({\mathbf x},\delta )\) and are continuous at \(\mathbf x\) then writing \begin {align*} f_{i}({\mathbf x}+{\mathbf h})=f_{i}({\mathbf x})& +\sum _{j=1}^{m}f_{i,j}({\mathbf x})h_{j}+ \sum _{j=1}^{m}\sum _{k=1}^{m}f_{i,jk}({\mathbf x})h_{j}h_{k}\\ &+\dots +\text {sum up to $n$th powers}+ \epsilon _{i}(\mathbf {h})\|\mathbf {h}\|^{n} \end {align*}

we have \(\|{\boldsymbol \epsilon }(\mathbf {h})\|\rightarrow 0\) as \(\|\mathbf {h}\|\rightarrow 0\).

Lemma 9.1 (Key integration property). If \(f:[a,b]\rightarrow {\mathbb R}\) is bounded and \(\mathcal {D}_{1}\) and \(\mathcal {D}_{2}\) are two dissections then

\begin{equation*}
S(f,\mathcal{D}_{1})\geq S(f,\mathcal{D}_{1}\cup\mathcal{D}_{2})
\geq
s(f,\mathcal{D}_{1}\cup\mathcal{D}_{2})\geq s(f,\mathcal{D}_{2}).
\tag*{$\bigstar$}
\end{equation*}

Lemma 9.2. If \(f:[a,b]\rightarrow {\mathbb R}\) is bounded then \(I^{*}(f)\geq I_{*}(f)\).

Lemma 9.3. (i) A bounded function \(f:[a,b]\rightarrow {\mathbb R}\) is Riemann integrable if and only if given any \(\epsilon >0\) we can find a dissection \(\mathcal D\) with \[S(f,{\mathcal D})-s(f,{\mathcal D})<\epsilon .\]

(ii) A bounded function \(f:[a,b]\rightarrow {\mathbb R}\) is Riemann integrable with integral \(I\) if and only if given any \(\epsilon >0\) we can find a dissection \(\mathcal D\) with \[S(f,{\mathcal D})-s(f,{\mathcal D})<\epsilon , \ \text {and}\ S(f,{\mathcal D})\geq I\geq s(f,{\mathcal D}).\]

Theorem 9.4. (i) Any monotonic (that is any increasing or decreasing) function \(f:[a,b]\rightarrow {\mathbb R}\) is Riemann integrable.

(ii) Any continuous function \(f:[a,b]\rightarrow {\mathbb R}\) is Riemann integrable.

Example 9.5. If \(f:[0,1]\rightarrow {\mathbb R}\) is given by \begin {align*} f(x)=0&\qquad \text {when $x$ is rational,}\\ f(x)=1&\qquad \text {when $x$ is irrational,} \end {align*}

then \(f\) is not Riemann integrable.

Lemma 9.6. Suppose that \(a\leq c\leq b\) and \(f:[a,b]\rightarrow {\mathbb R}\). If \(f|_{[a,c]}\in {\mathcal R}[a,c]\) and \(f|_{[c,b]}\in {\mathcal R}[c,b]\) then \(f\in {\mathcal R}[a,b]\) and \[\int _{a}^{b}f(x)\,dx=\int _{a}^{c}f(x)\,dx +\int _{c}^{b}f(x)\,dx .\]

Lemma 9.7. If \(\lambda ,\mu \in {\mathbb R}\) and \(f,g:[a,b]\rightarrow {\mathbb R}\) are Riemann integrable then so is \(\lambda f+\mu g\) and \[\int _{a}^{b}\lambda f(x)+\mu g(x)\,dx =\lambda \int _{a}^{b}f(x)\,dx+\mu \int _{a}^{b}g(x)\,dx.\]

Lemma 9.8. If \(f,g:[a,b]\rightarrow {\mathbb R}\) are Riemann integrable and \(f(x)\geq g(x)\) for all \(x\in [a,b]\) then \[\int _{a}^{b}f(x)\geq \int _{a}^{b}g(x)\,dx.\]

Lemma 9.9. If \(f:[a,b]\rightarrow {\mathbb R}\) is Riemann integrable then so is \(|f|\).

Lemma 9.10. (i) If \(f:[a,b]\rightarrow {\mathbb R}\) is Riemann integrable then so is \(f^{2}\).

(ii) If \(f,g:[a,b]\rightarrow {\mathbb R}\) are Riemann integrable then so is \(fg\).

Theorem 9.11 (Fundamental theorem of the calculus). Suppose that \(u\in (a,b)\) and \(f:(a,b)\rightarrow {\mathbb R}\) is a Riemann integrable function function which is continuous at some \(t\in (a,b)\). If we set \[F(s)=\int _{u}^{s}f(x)\,dx\] then \(F\) is differentiable at \(t\) and \(F'(t)=f(t)\).

Theorem 9.12. Suppose that \(f:(a,b)\rightarrow {\mathbb R}\) is continuous, that \(u\in (a,b)\) and \(c\in {\mathbb R}\). Then there is a unique solution to the differential equation \(g'(t)=f(t)\) \([t\in (a,b)]\) such that \(g(u)=c\).

Theorem 9.13 (Differentiation under the integral). Suppose \(g:[a,b]\times [c,d]\) is continuous and that the partial derivative \(g_{2}\) exists and is continuous. Then writing \(G(y)=\int _{a}^{b}g(x,y)\,dx\) we have \(G\) differentiable on \((c,d)\) with \[G'(y)=\int _{a}^{b}g_{,2}(x,y)\,dx.\]

Lemma 9.14.   Suppose that \(g:{\mathbb R}^{2}\rightarrow {\mathbb R}\) is continuous and has continuous partial derivative \(g_{,2}\).

(i) If we set \(G(x,y)=\int _{0}^{x}g(u,y)\,du\) then \(G\) is differentiable with partial derivatives \[G_{,1}(x,y)=g(x,y)\ \text {and} \ G_{,2}(x,y)=\int _{0}^{x}g_{,2}(u,y)\,du.\]

(ii) If we set \(F(t)=\int _{0}^{t}g(u,t)\,du\) then \(F\) is differentiable with derivative \[F'(t)=g(t,t)+\int _{0}^{t}g_{,2}(u,t)\,du.\]

Definition 10.1. If \(f:[a,\infty )\rightarrow {\mathbb R}\) is such that \(f|_{[a,X]}\in {\mathcal R}[a,X]\) for each \(X>a\) and \(\int _{a}^{X}f(x)\,dx\rightarrow L\) as \(X\rightarrow \infty \) then we say that \(\int _{a}^{\infty }f(x)\,dx\) exists with value \(L\).

Lemma 10.2. Suppose \(f:[a,\infty )\rightarrow {\mathbb R}\) is such that \(f|_{[a,X]}\in {\mathcal R}[a,X]\) for each \(X>a\). If \(f(x)\geq 0\) for all \(x\) then \(\int _{a}^{\infty }f(x)\,dx\) exists if and only if there exists a \(K\) such that \(\int _{a}^{X}f(x)\,dx\leq K\) for all \(X\).

Lemma 10.3. Suppose \(f:[a,\infty )\rightarrow {\mathbb R}\) is such that \(f|_{[a,X]}\in {\mathcal R}[a,X]\) for each \(X>a\). If \(\int _{a}^{\infty }|f(x)|\,dx\) exists then \(\int _{a}^{\infty }f(x)\,dx\) exists.

Example 10.4. If \(\lambda ,\mu >0\) then \[\int _{-\mu R}^{\lambda R}\frac {x}{1+x^{2}}\,dx \rightarrow \log \frac {\lambda }{\mu }\] as \(R\rightarrow \infty \).

Definition 10.5. If \(f:{\mathbb R}\rightarrow {\mathbb R}\) is such that \(f|_{[-X,Y]}\in {\mathcal R}[X,Y]\) for each \(X,Y>0\) \(\int _{-\infty }^{\infty }f(x)\,dx\) exists with value \(L\) if and only if the following condition holds. Given \(\epsilon >0\) we can find an \(X_{0}(\epsilon )>0\) such that \[\left |\int _{-X}^{Y}f(x)\,dx-L\right |<\epsilon .\] for all \(X,Y>X_{0}(\epsilon )\).

Definition 10.6. If \(f:[a,b]\rightarrow {\mathbb R}^{m}\) is such that \(f_{j}:[a,b]\rightarrow {\mathbb R}\) is Riemann integrable for each \(j\) then \(\int _{a}^{b}f(x)\,dx=\mathbf {y}\) where \({\mathbf y}\in {\mathbb R}^{m}\) and \[y_{j}=\int _{a}^{b}f_{j}(x)\,dx.\]

Lemma 10.7. If \(\alpha :{\mathbb R}^{m}\rightarrow {\mathbb R}^{p}\) is linear and \(f:[a,b]\rightarrow {\mathbb R}^{m}\) is Riemann integrable then so is \(\alpha \circ f\) and \[\int _{a}^{b}\alpha (f(x))\,dx=\alpha \left ( \int _{a}^{b}f(x)\,dx\right ).\]

Theorem 10.8. If \(f:[a,b]\rightarrow {\mathbb R}^{m}\) is Riemann integrable then \[\left \|\int _{a}^{b}f(x)\,dx\right \| \leq (b-a)\sup _{x\in [a,b]}\|f(x)\|.\]

Exercise 10.9. Show that the collection \(\mathcal R\) of Riemann integrable functions \(f:[a,b]\rightarrow {\mathbb R}^{m}\) form a real vector space. If we write \[Tf=\int _{a}^{b}f(x)\,dx\] show that \(T:{\mathcal R}\rightarrow {\mathbb R}\) is a linear map and \(|Tf|\leq (b-a)\|f\|\).

Definition 11.1. We say that \((X,d)\) is a metric space if \(X\) is a set and \(d:X^{2}\rightarrow \mathbb {R}\) is a function with the following properties:-

(i) \(d(x,y)\geq 0\) for all \(x,y\in X\).

(ii) \(d(x,y)=0\) if and only if \(x=y\).

(iii) \(d(x,y)=d(y,x)\) for all \(x,y\in X\).

(iv) (The triangle inequality) \(d(x,z)\leq d(x,y)+d(y,z)\)

Lemma 11.2.   If \(d({\mathbf x},{\mathbf y})=\|{\mathbf x}-{\mathbf y}\|\) then \(({\mathbb R}^{m},d)\) is a metric space.

Example 11.3. If \(d({\mathbf x},{\mathbf y})=\|{\mathbf x}\|+\|{\mathbf y}\|\) when \(\mathbf {x}\neq \mathbf {y}\) and \(d({\mathbf x},{\mathbf x})=0\) then \(({\mathbb R}^{m},d)\) is a metric space.

Example 11.4. If \(\mathbf x\) and \(\mathbf y\) are linearly dependent (in more geometrical language) if \(\mathbf x\), \(\mathbf y\) and \(\mathbf 0\) are on the same straight line) then set \[d({\mathbf x},{\mathbf y})=\|{\mathbf x}-{\mathbf y}\|.\] Otherwise, set \[d({\mathbf x},{\mathbf y})=\|{\mathbf x}\|+\|{\mathbf y}\|.\] With this definition, \(({\mathbb R}^{m},d)\) is a metric space.

Lemma 11.5 (Hamming metric). Let \(M\) be the space of sequences \[{\mathbf x}=(x_{1},x_{2},\dots ,x_{n})\] with \(x_{j}=0\) or \(x_{j}=1\). We call \(M\) the space of messages of length \(n\). If \(\mathbf x\) and \(\mathbf y\) are messages we define \[d({\mathbf x},{\mathbf y})=\sum _{j=1}^{n}|x_{j}-y_{j}|\] to be the Hamming distance between the two messages. \((M,d)\) is a metric space.

Definition 11.6. Let \((X,d)\) be a metric space. If \(a_{n}\in X\) for each \(n\geq 1\) and \(a\in X\) then we say that \(a_{n}\rightarrow a\) if given \(\epsilon >0\) we can find an \(n_{0}(\epsilon )\) such that \[d(a_{n},a)<\epsilon \ \text {for all $n\geq n_{0}(\epsilon )$}.\]

Lemma 11.7. Let \((X,d)\) be a metric space.

(i) The limit is unique. That is, if \(a_{n}\rightarrow a\) and \(a_{n}\rightarrow b\) as \(n\rightarrow \infty \) then \(a=b\).

(ii) If \(a_{n}\rightarrow a\) as \(n\rightarrow \infty \) and \(n(1)<n(2)<n(3)\ldots \) then \(a_{n(j)}\rightarrow a\) as \(j\rightarrow \infty \).

(iii) If \(a_{n}=c\) for all \(n\) then \(a_{n}\rightarrow c\) as \(n\rightarrow \infty \).

Definition 11.8. Let \((X,d)\) be metric space. A set \(F\subseteq X\) is closed if whenever \(x_{n}\in F\) for each \(n\) and \(x_{n}\rightarrow x\) as \(n\rightarrow \infty \) then \(x\in F\).

Definition 11.9. Let \((X,d)\) be metric space A set \(U\subseteq X\) is open if whenever \(x\in U\) there exists an \(\epsilon >0\) such that whenever \(d(x,y)<\epsilon \) we have \(y\in U\).

Example 11.10. Let \((X,d)\) be metric space. Let \(x\in X\) and \(r>0\).

(i) The set \(B(x,r) =\{y\in X:d(x,y)<r\}\) is open.

(ii) The set \(\bar {B}(x,r) =\{y\in X:d(x,y)\leq r\}\) is closed.

Lemma 11.11. Let \((X,d)\) be a metric space. A subset \(A\) of \(X\) is open if and only if each point of \(A\) is the centre of an open ball lying entirely within \(A\).

Definition 11.12. The set \(N\) is a neighbourhood of the point \(x\) if we can find an \(r>0\) such that \(B(x,r)\subseteq N\).

Lemma 11.13. Let \((X,d)\) be a metric space. A subset \(A\) of \(X\) is open if and only if its complement \(X\setminus A\) is closed.

Lemma 11.14. Let \((X,d)\) be a metric space. Consider the collection \(\tau \) of open sets in \(X\).

(i) \(\emptyset \in \tau \), \(X\in \tau \).

(ii) If \(U_{\alpha }\in \tau \) for all \(\alpha \in A\) then \(\bigcup _{\alpha \in A} U_{\alpha }\in \tau \).

(iii) If \(U_{1},U_{2},\dots ,U_{n}\in \tau \) then \(\bigcap _{j=1}^{n}U_{j}\in \tau \).

Lemma 11.15. Let \((X,d)\) be a metric space. Consider the collection \(\mathcal {F}\) of closed sets in \(X\).

(i) \(\emptyset \in \mathcal {F}\), \(X\in \mathcal {F}\).

(ii) If \(F_{\alpha }\in \mathcal {F}\) for all \(\alpha \in A\) then \(\bigcap _{\alpha \in A} F_{\alpha }\in \mathcal {F}\).

(iii) If \(F_{1},F_{2},\dots ,F_{n}\in \mathcal {F}\) then \(\bigcup _{j=1}^{n}F_{j}\in \mathcal {F}\).

Definition 11.16. Let \((X,d)\) and \((Z,\rho )\) be metric spaces. We say that a function \(f:X \rightarrow Z\) is continuous at some point \(x\in X\) if given \(\epsilon >0\) we can find a \(\delta (\epsilon ,x)>0\) such that if \(y\in X\) and \(d(x,y)<\delta (\epsilon ,x)\) we have \[\rho [(f(x),f(y))<\epsilon .\] If \(f\) is continuous at every point \(x\in X\) we say that \(f\) is a continuous function on \(X\).

Lemma 11.17. Let \((X,d)\) be a metric space and \(E\subseteq X\). Let \(d_{E}:E^{2}\rightarrow {\mathbb R}\) be given by \(d_{E}(u,v)=d(u,v)\) whenever \(u,v\in E\). Then \((E,d_{E})\) is a metric space.

Lemma 11.18. Let \((X,d)\) and \((Z,\rho )\) be metric spaces and suppose that the function \(f:X\rightarrow Z\) is continuous at \(x\in X\). Then if \(x_{n}\in X\) and \(x_{n}\rightarrow x\) it follows that \(f(x_{n})\rightarrow f(x)\).

Lemma 11.19. Let \((X,d)\) and \((Z,\rho )\) be metric spaces. The function \(f:X\rightarrow Z\) is continuous if and only if \(f^{-1}(O)\) is open whenever \(O\) is open.

Lemma 11.20. Let \((X,d)\) \((Y,\theta )\) and \((Z,\rho )\) be metric spaces. If \(f:X\rightarrow Y\) and \(g:Y\rightarrow Z\) are continuous then so is their composition \(g\circ f\).

Statement 12.1. (i) \(d(A,B)\geq 0\) for all \(A\) and \(B\).

(ii) \(d(A,B)=0\) if and only if \(A=B\).

(iii) \(d(A,B)=d(B,A)\) for all \(A\) and \(B\).

(iv) \(d(A,C)\leq d(A,B)+d(B,C)\) for all \(A\), \(B\) and \(C\).

Statement 12.2. The set \(\mathcal {H}\) of Möbius maps \(T\) such that \(T|_{H}:H\rightarrow H\) is bijective is a subgroup of the group \(\mathcal {M}\) of all Möbius transformations. The subgroup \(\mathcal {H}\) is generated by the transformations \(T_{a}\) with \(a\in {\mathbb R}\), \(D_{\lambda }\) with \(\lambda >0\) and \(J\) where \begin {align*} T_{a}=&a+z\\ D_{\lambda }(z)=&\lambda z\\ J(z)=&-z^{-1} \end {align*}

Statement 12.3. Suppose \(g:H\rightarrow {\mathbb R}\) is well behaved. Then \[\int _{T(\Gamma )}g(x,y)ds=\int _{\Gamma }g(x,y)ds\] for all nice paths \(\Gamma \) in \(H\) and all \(T\in {\mathcal H}\) if and only if \(g(x,y)=Ay^{-1}\) for some \(A>0\).

Statement 12.4. The metric just described is invariant under Möbius transformation.

Statement 12.5. The geodesics for the metric just described are arcs of circles with centres on the real axis and straight lines which cut the real axis at right angles.

Lemma 12.6. Let \(G\) be a group and \(A\) a subset of \(G\) generating \(G\). Write \(A^{-1}=\{a^{-1}:a\in A\}\). If we set \[d(x,y)=\min \{n: \text {we can find $a_{1},a_{2},\dots ,a_{n}\in A\cup A^{-1}$ with $a_{1}a_{2}\dots a_{n}=xy^{-1}$}\},\] then \(d\) is a metric on \(G\) with \(d(xz,yz)=d(x,y)\) for all \(x,y,z\in G\). (Thus \(d\) is a right translation invariant metric.)

Definition 13.1. If \((X,d)\) is a metric space we say that a sequence of points \(x_{n}\in X\) is Cauchy if given any \(\epsilon >0\) we can find \(n_{0}(\epsilon )\) such that \(d(x_{p},x_{q})<\epsilon \) for all \(p,q\geq n_{0}(\epsilon )\).

Definition 13.2. A metric space \((X,d)\) is complete if every Cauchy sequence converges.

Lemma 13.3. Let \(X\) be any set. If we define \(d:X^{2}\rightarrow \mathbb {R}\) by \begin {align*} d(x,y)=&1\qquad \text {if $x\neq y$}\\ d(x,x)=&0 \end {align*}

then \((X,d)\) is a metric space.

Lemma 13.4. Let \((X,d)\) be a space with the discrete metric. Then \((X,d)\) is complete.

Suppose \(X\) is infinite. Then although \(X\) is bounded (if \(x_{0}\in X\) then \(\bar {B}(x_{0},1)=X\)) we can find a sequence \(x_{n}\) such that no subsequence converges.

Lemma 13.5. If \((X,d)\) is a space with the discrete metric then every subset of \(X\) is both open and closed.

Exercise 13.6. We say that a metric space \((X,d)\) has no isolated points if given \(y\in X\) and \(\epsilon >0\) we can find an \(x\in X\) such that \(0<d(x,y)<\epsilon \). Show by the methods of Exercise 3.14 that a non-empty metric space with no isolated points is uncountable.

Give an example of an infinite countable metric space. Give an example of an uncountable metric space all of whose points are isolated.

Lemma 13.7. Let \((X,d)\) be a complete metric space. If \(E\) is a subset of \(X\) and we define \(d_{E}:E^{2}\rightarrow {\mathbb R}\) by \(d_{E}(u,v)=d(u,v)\) whenever \(u,v\in E\) then \((E,d_{E})\) is complete if and only if \(E\) is closed in \((X,d)\).

Example 13.8. Consider \(C([-1,1])\) the set of continuous functions \(f:[-1,1]\rightarrow {\mathbb R}\). If we set \[d(f,g)=\|f-g\|_{1}=\int _{-1}^{1}|f(x)-g(x)|\,dx\] then \(d\) is a metric. Let

\begin{alignat*}{2}
f_{n}(x)&=-1&&\qquad\text{for $-1\leq x\leq -1/n$}\\
f_{n}(x)&=nx&&\qquad\text{for $-1/n\leq x\leq 1/n$}\\
f_{n}(x)&=1&&\qquad\text{for $1/n\leq x\leq 1$}.
\end{alignat*}

The sequence \(f_{n}\) is Cauchy but has no limit. Thus \((C([-1,1]),d)\) is not complete.

Exercise 13.9. Consider \({\mathcal R}([-1,1])\) the set of Riemann integrable functions functions \(f:[-1,1]\rightarrow {\mathbb R}\). If we set \[d(f,g)=\|f-g\|_{1}=\int _{-1}^{1}|f(x)-g(x)|\,dx\] show that \(d\) satisfies conditions (i), (iii) and (iv) of the definition of a metric (see Definition 11.1) but give an example to show that condition (ii) may fail. We could now stop since \(d\) is not a metric but this would be to miss the interesting part.

In many ways, condition (ii) is the least important part of the definition of a metric. We call something satisfying all the other conditions a pseudo-metric. We can define limits for a pseudo-metrics in the same way as for metrics.

If \((X,\rho )\) is a pseudo-metric space show that if \(x_{n}\rightarrow x\) then \(x_{n}\rightarrow y\) if and only if \(\rho (x,y)=0\).

In the same way we can define Cauchy sequences. A pseudo-metric space is called complete if every Cauchy sequence converges. Our object is to show that \(({\mathcal R}([-1,1]),d)\) is not complete.

(i) Set

\begin{alignat*}{2}
\Delta_{n}(x)=&0&&\qquad\text{if $2^{-3n}\leq |x|$}\\
\Delta_{n}(x)=&1-2^{3n}|x|&&\qquad\text{if $|x|\leq 2^{-3n}$.}
\end{alignat*}

and \(g_{n}(t)=\sum _{r=-2^{n}}^{2^{n}}\Delta _{n}(t-r2^{-n})\) for all \(-1\leq t\leq 1\). Sketch \(g_{n}\) and show that \(\|g_{n}\|_{1}\leq 2^{-2n+1}\).

(ii) If \(f_{n}(x)=\max \{f_{1}(x),f_{2}(x),\dots ,f_{n}(x)\}\) explain briefly why \(f_{n}\) is a continuous function on \([-1,1]\) with \(0\leq f_{n}(x)\leq 1\). Show that \(\|f_{n+1}-f_{n}\|\leq 2^{-2n+3}\) and deduce that \(f_{n}\) is a Cauchy sequence in \(({\mathcal R}([-1,1]),d)\).

(iii) Suppose if possible that there exists an \(f\in {\mathcal R}([-1,1])\) such that \(\|f_{n}-f\|\rightarrow 0\) as \(n\rightarrow \infty \). Show that \(\|f\|\leq 2/3\).

(iv) We now use the notation of Section 9. Explain why there must exist a dissection \(\mathcal {D}\) of \([-1,1]\) such that \(S(f,\mathcal {D})\leq 1\). Deduce that we can find \(a\) and \(b\) with \(-1\leq a <b\leq 1\) such that \(f(x)\leq 1/2\) for all \(x\in [a,b]\).

(v) Show that \(\int _{a}^{b}|f(x)-f_{n}(x)|\,dx\nrightarrow 0\) and deduce that \(\|f-f_{n}\|_{1}\nrightarrow 0\). This contradiction shows that the sequence \(f_{n}\) does not converge and \(({\mathcal R}([-1,1]),d)\) is not complete.

The reader who has got this far can feel well pleased with herself but by working still harder we can extract a little more juice from this example and show that \(f_{n}\) converges pointwise to a function which is not Riemann integrable.

(v) Show that, for each \(x\), \(f_{n}(x)\) is a bounded increasing function. Deduce that \(f_{n}(x)\) tends to limit \(F(x)\), say, for each \(x\). Show that \(I^{*}(F)=2\).

(vi) Let \(Z_{n}=f_{n}^{-1}(0)\). Explain why \(Z_{n}\) is closed and \(Z_{1}\supseteq Z_{2}\supseteq \dots \). If \(\mathcal {D}\) is a fixed dissection of \([-1,1]\) show that for each \(n\) we can find a collection of \(\mathcal {I}_{n}\) of intervals \(I=[x_{j},x_{j-1}]\) with end points successive points of the dissection and of total length greater than \(1\) such that \(I\cap Z_{n}\neq \emptyset \).

(vii) Let \(\mathcal {I}=\bigcap _{n=1}^{\infty }\mathcal {I}_{n}\). Show that \(\mathcal {I}\) is a collection of intervals \(I=[x_{j},x_{j-1}]\) with end points successive points of the dissection \(\mathcal {D}\) and with total length greater than \(1\) such that \(I\cap Z_{n}\neq \emptyset \) for each \(n\).

(viii) Use Theorem 5.17 to show that \(I\cap \bigcap _{n=1}^{\infty }Z_{n}\neq \emptyset \) whenever \(I\in mathcal{I}\). Deduce that \(s(F,\mathcal {D})\leq 1\).

(ix) Conclude that \(F\) is not Riemann integrable.

Definition 14.1. If \(E\) is a non-empty set we write \(\mathcal {B}(E)\) for the set of bounded functions \(f:E\rightarrow {\mathbb F}\). The uniform norm \(\|\ \|_{\infty }\) on \(\mathcal {B}\) is defined by \(\|f\|_{\infty }=\sup _{x\in E}|f(x)|\).

Lemma 14.2. If we use the standard operations \(\mathcal {B}(E)\) is a vector space over \(\mathbb {F}\). If \(f,g,h\in \mathcal {B}(E)\) the following results are true.

(i) \(\|f\|_{\infty }\geq 0\),

(ii) If \(\|f\|_{\infty }= 0\) then \(f=0\),

(iii) If \(\lambda \in {\mathbb F}\) then \(\|\lambda f\|_{\infty }=|\lambda |\|f\|_{\infty }\).

(iv) (The triangle inequality) \(\|f+g\|_{\infty }\leq \|f\|_{\infty }+\|g\|_{\infty }\).

(v) The pointwise product \(fg\in \mathcal {B}(E)\) (where \(fg(x)=f(x)g(x)\)) and \(\|fg\|_{\infty }\leq \|f\|_{\infty }\|g\|_{\infty }\).

Theorem 14.3. The uniform metric on \(\mathcal {B}(E)\) is complete.

Definition 14.4. If \((E,d)\) is a non-empty metric space we write \(\mathcal {C}(E)\) for the set of bounded continuous functions \(f:E\rightarrow {\mathbb F}\).

Lemma 14.5. If \((E,d)\) is a non-empty metric space then \(\mathcal {C}(E)\) is a vector subspace of \(\mathcal {B}(E)\). Further if \(f,g\in \mathcal {C}(E)\) the pointwise product \(fg\in \mathcal {C}(E)\).

Theorem 14.6. If \((E,d)\) is a non-empty metric space then \(\mathcal {C}(E)\) is a closed subset of \(\mathcal {B}(E)\) under the uniform metric.

Definition 14.7 (Uniform convergence). If \(E\) is a non-empty set and \(f_{n}:E\rightarrow {\mathbb F}\) and \(f:E\rightarrow {\mathbb F}\) are functions we say that \(f_{n}\) converges uniformly to \(f\) as \(n\rightarrow \infty \) if, given any \(\epsilon >0\) we can find an \(n_{0}(\epsilon )\) such that \(|f_{n}(x)-f(x)|<\epsilon \) for all \(x\in E\) and all \(n\geq n_{0}(\epsilon )\).

Theorem 14.8. If \((E,d)\) is a non-empty metric space and \(f_{n}:E\rightarrow {\mathbb F}\) form a sequence of continuous functions tending uniformly to \(f\) then \(f\) is continuous.

Theorem 14.9. If \((E,d)\) is a non-empty metric space then the the uniform metric on \(\mathcal {C}(E)\) is complete.

Theorem 14.10 (General principle of uniform convergence). Suppose that \((E,d)\) is a non-empty metric space and \(f_{n}:E\rightarrow {\mathbb F}\) is a continuous function \([n\geq 1]\). The sequence \(f_{n}\) converge uniformly to a continuous function \(f\) if and only if given any \(\epsilon >0\) we can find an \(n_{0}(\epsilon )\) such that \(|f_{n}(x)-f_{m}(x)|<\epsilon \) for all \(n,m\geq n_{0}(\epsilon )\).

Example 14.11 (The witch’s hat). Define \(f_{n}:[0,2]\rightarrow {\mathbb R}\) by

\begin{alignat*}{2}
f_{n}(x)=&1-n|x-n^{-1}|&&\qquad\text{for $|x-n^{-1}|\leq n^{-1}$,}\\
f_{n}(x)=&0&&\qquad\text{otherwise.}
\end{alignat*}

Then the \(f_{n}\) are continuous functions such that \(f_{n}(x)\rightarrow 0\) as \(n\rightarrow \infty \) for each \(x\) but \(f_{n}\nrightarrow 0\) uniformly.

Example 14.12. Define \(f_{n}:[0,1]\rightarrow {\mathbb R}\) by \(f_{n}(x)=x^{n}\). Then \(f_{n}(x)\rightarrow f(x)\) as \(n\rightarrow \infty \) where \(f(x)=0\) for \(0\leq x<1\) but \(f(1)=1\).

Theorem 14.13. Let \(f_{n}\in C([a,b])\). If \(f_{n}\rightarrow f\) uniformly then \(f\in C([a,b])\) and \[\int _{a}^{b}f_{n}(x)\,dx\rightarrow \int _{a}^{b}f(x)\,dx.\]

Example 14.14 (The tall witch’s hat). Define \(f_{n}:[0,2]\rightarrow {\mathbb R}\) by

\begin{alignat*}{2}
f_{n}(x)=&n(1-n|x-n^{-1}|)&&\qquad
\text{for $|x-n^{-1}|\leq n^{-1}$,}\\
f_{n}(x)=&0&&\qquad\text{otherwise.}
\end{alignat*}

Then the \(f_{n}\) are continuous functions such that \(f_{n}(x)\rightarrow 0\) as \(n\rightarrow \infty \) but \[\int _{0}^{2}f_{n}(x)\,dx\nrightarrow 0\] as \(n\rightarrow \infty \).

Example 14.15 (Escape to infinity). Define \(f_{n}:{\mathbb R}\rightarrow {\mathbb R}\) by

\begin{alignat*}{2}
f_{n}(x)=&1-n^{-1}|x|&&\qquad
\text{for $|x|\leq n$,}\\
f_{n}(x)=&0&&\qquad\text{otherwise.}
\end{alignat*}

Then the \(f_{n}\) are continuous functions such that \(f_{n}(x)\rightarrow 0\) uniformly as \(n\rightarrow \infty \) but \[\int _{-\infty }^{\infty }f_{n}(x)\,dx\nrightarrow 0\] as \(n\rightarrow \infty \).

Theorem 14.16. Suppose that \(f_{n}:[a,b]\rightarrow {\mathbb R}\) is differentiable on \([a,b]\) with continuous derivative \(f_{n}'\) (we take the one-sided derivative at end points if necessary). Suppose that \(f_{n}(x)\rightarrow f(x)\) as \(n\rightarrow \infty \) for each \(x\in [a,b]\) and suppose that \(f_{n}'\) converges uniformly to a limit \(F\) on \([a,b]\). Then \(f\) is differentiable with derivative \(F\).

Theorem 14.17 (Term by term integration). Let \(g_{j}:[a,b]\rightarrow {\mathbb R}\) be continuous. If \(\sum _{j=1}^{n}g_{j}\) converges uniformly as \(n\rightarrow \infty \) then \[\int _{a}^{b}\sum _{j=1}^{\infty }g_{j}(x)\,dx =\sum _{j=1}^{\infty }\int _{a}^{b}g_{j}(x)\,dx.\]

Theorem 14.18 (Term by term differentiation). Let \(g_{j}:[a,b]\rightarrow {\mathbb R}\) be differentiable with continuous derivative. If \(\sum _{j=1}^{n}g_{j}(x)\) converges for each \(x\) and \(\sum _{j=1}^{n}g_{j}'\) converges uniformly as \(n\rightarrow \infty \) then \(\sum _{j=1}^{\infty }g_{j}\) is differentiable and \[\frac {d\ }{dx}\left (\sum _{j=1}^{\infty }g_{j}(x)\right ) =\sum _{j=1}^{\infty }g_{j}(x).\]

Theorem 14.19 (Differentiation under an infinite integral). Suppose \(g:[0,\infty ]\times [c,d]\) is continuous and that the partial derivative \(g_{,2}\) exists and is continuous. Suppose further that that there exists a continuous function \(h:[0,\infty ]\times [c,d]\) with \(|g_{,2}(x,y)|\leq h(x)\) for all \((x,y)\) and such that \(\int _{0}^{\infty }h(x)\,dx\) exists and is finite. Then, if \(G(y)=\int _{0}^{\infty }g(x,y)\,dx\) exists for all \(y\in (c,d)\) we have \(G\) differentiable on \((c,d)\) with \[G'(y)=\int _{0}^{\infty }g_{,2}(x,y)\,dx.\]

Theorem 14.20. We work in \(\mathbb C\). Let \(\sum _{n=0}^{\infty }a_{n}z^{n}\) be a power series with radius of convergence \(R\). If \(0\leq r< R\) then \(\sum _{n=0}^{N}a_{n}z^{n}\) converges uniformly on \(\bar {B}(0,r)\) as \(N\rightarrow \infty \). However, it need not be true that \(\sum _{n=0}^{N}a_{n}z^{n}\) converges uniformly on \(B(0,R)\).

Theorem 15.1 (The contraction mapping theorem). Let \((X,d)\) be a non-empty complete metric space and \(T:X\rightarrow X\) a mapping such that there exists a \(K<1\) with \(d(Tx,Ty)\leq K d(x,y)\) for all \(x,y\in X\). Then there exists a unique \(x_{0}\in X\) such that \(Tx_{0}=x_{0}\).

Example 15.2. (i) If \(X\) is the unit circle centre \(\mathbf 0\) in the plane, \(d\) Euclidean distance and \(T\) a non-trivial rotation then \((X,d)\) is a complete metric space and \(d(T{\mathbf x},T{\mathbf y})=d({\mathbf x},{\mathbf y})\) for all \({\mathbf x},{\mathbf y}\in X\) but \(T\) has no fixed point.

(ii) If \(X={\mathbb R}\), \(d\) is Euclidean distance and \[Tx=1+x+g(x)\] for some suitably chosen \(g\) then \((X,d)\) is a complete metric space and \(d(Tx,Ty)< d(x,y)\) for all \(x,y\in X\) but \(T\) has no fixed point.

Lemma 15.3. If \(f:{\mathbb R}^{2}\rightarrow {\mathbb R}\) is continuous, \(x_{0},y_{0}\in {\mathbb R}\) and \(\delta >0\) then the following two statements are equivalent.

(A) The function \(y:(x_{0}-\delta ,x_{0}+\delta )\rightarrow {\mathbb R}\) is differentiable and satisfies the equation \(y'(x)=f(x,y(x))\) for all \(x\in (x_{0}-\delta ,x_{0}+\delta )\) together with the boundary condition \(y(x_{0})=y_{0}\).

(B) The function \(y:(x_{0}-\delta ,x_{0}+\delta )\rightarrow {\mathbb R}\) is continuous and satisfies the condition \[y(x)=y_{0}+\int _{x_{0}}^{x}f(u,y(u))\,du\] for all \(x\in (x_{0}-\delta ,x_{0}+\delta )\).

Theorem 15.4.   Suppose \(f:{\mathbb R}^{2}\rightarrow {\mathbb R}\) is continuous, \(x_{0},y_{0}\in {\mathbb R}\) and \(\delta >0\). Suppose further that there exists a \(K>0\) such that \(K\delta <1\) and

\begin{equation*}
|f(x,u)-f(x,v)|\leq K|u-v| \tag*{$\bigstar$}
\end{equation*}

for all \(u,v\in (x_{0}-\delta ,x_{0}+\delta )\). Then there exists a unique \(y:(x_{0}-\delta ,x_{0}+\delta )\rightarrow {\mathbb R}\) is continuous and satisfies the condition \[y(x)=y_{0}+\int _{x_{0}}^{x}f(t,y(t))\,dt\] for all \(x\in (x_{0}-\delta ,x_{0}+\delta )\).

Example 15.5. There are an infinity of different solutions to the equation \[y'=3y^{2/3}\] with \(y(0)=0\).

Theorem 15.6. Suppose \(f:{\mathbb R}^{2}\rightarrow {\mathbb R}\) is a continuous function satisfying the following condition. There exists a \(K:[0,\infty )\rightarrow {\mathbb R}\) such that \[|f(x,u)-f(x,v)|\leq K(R) |u-v| \] whenever \(|x|,|u|,|v|\leq R\). Then given any \((x_{0},y_{0})\in {\mathbb R}^{2}\) we can find a \(\delta (x_{0},y_{0})>0\) such that there exists a unique differentiable function \(y:(x_{0}-\delta ,x_{0}+\delta )\rightarrow {\mathbb R}\) which satisfies the equation \(y'(x)=f(x,y(x))\) for all \(x\in (x_{0}-\delta ,x_{0}+\delta )\) together with the boundary condition \(y(x_{0})=y_{0}\).

Theorem 15.7. Suppose \(f:{\mathbb R}^{2}\rightarrow {\mathbb R}\) is a continuous function satisfying the following condition. There exists a \(K:[0,\infty )\rightarrow {\mathbb R}\) such that \[|f(x,u)-f(x,v)|\leq K(R) |u-v| \] for all \(|x|\leq R\) and all \(u\) and \(v\). Then given any \((x_{0},y_{0})\in {\mathbb R}^{2}\) we can find a unique \(y:{\mathbb R}\rightarrow {\mathbb R}\) which is is differentiable and satisfies the equation \(y'(x)=f(x,y(x))\) for all \(x\in {\mathbb R}\) together with the boundary condition \(y(x_{0})=y_{0}\)

Example 15.8. If \(f(x,y)=1+y^{2}\) then \[|f(x,u)-f(x,v)|\leq 2R |u-v| \] whenever \(|u|,|v|\leq R\). However, there does not exist a differentiable function \(y:{\mathbb R}\rightarrow {\mathbb R}\) with \(y'=f(x,y)\) for all \(x\).

Theorem 16.1. Suppose \(f:{\mathbb R}\times {\mathbb R}^{m} \rightarrow {\mathbb R}^{m}\) is a continuous function satisfying the following condition. There exists a \(K:[0,\infty )\rightarrow {\mathbb R}\) such that \[\|f(x,{\mathbf w})-f(x,{\mathbf z})\| \leq K(R) \|{\mathbf w}-{\mathbf z}\| \] for all \(|x|\leq R\) and all \(\mathbf {w}\) and \(\mathbf {z}\). Then given any \(x_{0}\in {\mathbb R}\) and \({\mathbf y}_{0}\in {\mathbb R}^{m}\) we can find a unique \(\mathbf {y}:{\mathbb R}\rightarrow {\mathbb R}^{m}\) which is is differentiable and satisfies the equation \[{\mathbf y}'(x)=f(x,{\mathbf y})\] for all \(x\in {\mathbb R}\) together with the boundary condition \(\mathbf {y}(x_{0})=\mathbf {y}_{0}\).

Lemma 16.2. (i) The Wronskian is differentiable with \[W'(x)=-a(x)W(x).\]

(ii) The Wronskian is never zero.

Theorem 16.3. If \(f:{\mathbb R}\rightarrow {\mathbb R}\) is continuous then \[y(x)=\int _{0}^{1}G(x,w)f(w)\,dw\] is twice differentiable and satisfies

\begin{equation*}
y’’+a(x)y’+b(x)y=f(x) \tag*{$(*)$}
\end{equation*}

together with the conditions \(y(0)=y(1)=0\).

Example 16.4. The equation \[y''+\pi ^{2}y=x\] has no solution satisfying \(y(0)=y(1)=0\).

Q 18.1. (We work with the same ideas as in Example 1.1.) (i) Find a differentiable function \(f:{\mathbb Q}\rightarrow {\mathbb Q}\) such that \(f'(x)=1\) for all \(x\in {\mathbb Q}\) but \(f(0)>f(1)\).

(ii) If we define \(g:{\mathbb Q}\rightarrow {\mathbb Q}\) by \(g(x)=(x^{2}-2)^{-2}\) show that \(g\) is continuous but unbounded on \(\{x\in {\mathbb Q}:|x|\leq 2\}\).

Q 18.2. In Question 18.3 below you are asked to prove Lemma 2.1. Write a paragraph on each of the following subjects.

(i) Would it have been better for the lecturer to prove Lemma 2.1 rather than leave it to you. Why?

(ii) What benefit, if any, do you expect to derive from doing Lemma 2.1? Give reasons.

(iii) What is the best way of approaching Question 18.1. For example should you look at your notes from last year before attacking the question? Should you look at your notes from last year after attacking the question?

Q 18.3.   Prove Lemma 2.1 (restated below for your convenience)

(i) The limit is unique. That is, if \(a_{n}\rightarrow a\) and \(a_{n}\rightarrow b\) as \(n\rightarrow \infty \) then \(a=b\).

(ii) If \(a_{n}\rightarrow a\) as \(n\rightarrow \infty \) and \(n(1)<n(2)<n(3)\ldots \) then \(a_{n(j)}\rightarrow a\) as \(j\rightarrow \infty \).

(iii) If \(a_{n}=c\) for all \(n\) then \(a_{n}\rightarrow c\) as \(n\rightarrow \infty \).

(iv) If \(a_{n}\rightarrow a\) and \(b_{n}\rightarrow b\) as \(n\rightarrow \infty \) then \(a_{n}+b_{n}\rightarrow a+b\).

(v) If \(a_{n}\rightarrow a\) and \(b_{n}\rightarrow b\) as \(n\rightarrow \infty \) then \(a_{n}b_{n}\rightarrow ab\).

(vi) If \(a_{n}\rightarrow a\) as \(n\rightarrow \infty \) and \(a_{n}\neq 0\) for each \(n\), \(a\neq 0\) then \(a_{n}^{-1}\rightarrow a^{-1}\).

(vii) If \(a_{n}\leq A\) for each \(n\) and \(a_{n}\rightarrow a\) as \(n\rightarrow \infty \) then \(a\leq A\).

Q 18.4. Prove Lemma 2.2 which states that a decreasing sequence of real numbers bounded below tends to a limit.

Q 18.5. (i) If \(a_{j}\) is a integer with \(0\leq a_{j}\leq 9\) show from the fundamental axiom that \[\sum _{j=1}^{\infty }a_{j}10^{-j}\] exists. Show that \(0\leq \sum _{j=1}^{\infty }a_{j}10^{-j}\leq 1\) carefully quoting any theorems that you use.

(ii) If \(0\leq x\leq 1\) show that we can find integers \(x_{j}\) with \(0\leq x_{j}\leq 9\) such that \[x=\sum _{j=1}^{\infty }x_{j}10^{-j}.\] What important deep result do you use?

(iii) If \(a_{j}\) and \(b_{j}\) are integers with \(0\leq a_{j},b_{j}\leq 9\) and \(a_{j}=b_{j}\) for \(j<N\), \(a_{N}>b_{N}\) show that \[\sum _{j=1}^{\infty }a_{j}10^{-j}\geq \sum _{j=1}^{\infty }b_{j}10^{-j}.\] Give the precise necessary and sufficient condition for equality and prove it.

Q 18.6. (i) Let us write \[S_{n}=\sum _{r=0}^{n}\frac {1}{r!}.\] Show, from first principles, that \(S_{n}\) converges to a limit (which, with the benefit of extra knowledge, we call \(e\)).

(ii) Show that, if \(n\geq 2\) and \(r\geq 0\) then \[\frac {n!}{(n+r)!}\leq \frac {1}{3^{r}}.\] Deduce carefully that, if \(m\geq n\geq 2\), \[0\leq n!(S_{m}-S_{n})\leq \frac {1}{2}.\] and that \[0<n!(e-S_{n})\leq \frac {1}{2}.\] Deduce that \(n!e\) is not an integer for any \(n\) and conclude that \(e\) is irrational.

(iii) Show similarly that \(\displaystyle \sum _{r=0}^{\infty }\frac {1}{(2r)!}\) is irrational.

Q 18.7. In this question you should feel free to use any results you know (provided you quote them correctly).

(i) By using the mean value theorem, show that \((1+x^{-1})^{x}\) is an increasing function of \(x\) for \(x\geq 0\).

(ii) Show that \((1+n^{-1})^{n}\) tends to limit \(L\) as \(n\rightarrow \infty \).

(iii) Show that, if \(n\geq N\), \[e\geq \left (1+\frac {1}{n}\right )^{n}\geq \sum _{r=0}^{N}\binom {n}{r}n^{-r}.\] Deduce that \[e\geq L\geq \sum _{r=0}^{N}\frac {1}{r!},\] and conclude that \[\left (1+\frac {1}{n}\right )^{n}\rightarrow e\] as \(n\rightarrow \infty \).

(iv) Assuming the Taylor expansion for \(e^{t}\) show that \[\left (1+\frac {t}{n}\right )^{n}\rightarrow e^{t}\] as \(n\rightarrow \infty \) for all \(t>0\).

(v) Show that \[1-\left (1+\frac {t^{2}}{n^{2}}\right )^{n}\rightarrow 0\] and deduce that \[1-\left (1-\frac {t^{2}}{n^{2}}\right )^{n}\rightarrow 0\] as \(n\rightarrow \infty \). Hence deduce that \[\left (1-\frac {t}{n}\right )^{n}\rightarrow e^{-t}\] as \(n\rightarrow \infty \) for all \(t>0\).

(vi) Conclude that \[\left (1+\frac {t}{n}\right )^{n}\rightarrow e^{t}\] as \(n\rightarrow \infty \) for all real \(t\).

Q 18.8. (a) Prove the statements made in Example 3.9.

(i) The sum \(\sum _{n=1}^{\infty }n^{-2}z^{n}\) has radius of convergence \(1\) and converges for all \(|z|=1\).

(ii) The sum \(\sum _{n=0}^{\infty }z^{n}\) has radius of convergence \(1\) and diverges for all \(|z|=1\).

(iii) The sum \(\sum _{n=1}^{\infty }n^{-1}z^{n}\) has radius of convergence \(1\), diverges for \(z=1\) and converges for \(z=-1\).

(b) Suppose that \(\sum _{j=0}^{\infty }a_{j}z^{j}\) has radius of convergence \(R\) and \(\sum _{j=0}^{\infty }b_{j}z^{j}\) has radius of convergence \(S\).

(i) Show that if \(R\neq S\) then \(\sum _{j=0}^{\infty }(a_{j}+b_{j})z^{j}\) has radius of convergence \(\min (R,S)\).

(ii) Show that if \(R=S\) then \(\sum _{j=0}^{\infty }(a_{j}+b_{j})z^{j}\) has radius of convergence at least \(R\).

(iii) If \(T\geq 1=R=S\) give an example where \(\sum _{j=0}^{\infty }(a_{j}+b_{j})z^{j}\) has radius of convergence \(T\).

Q 18.9. (i) Suppose that \(x_{n}\) is a bounded sequence of real numbers. Show that \(y_{n}=\sup _{m\geq n}x_{m}\) is a well defined bounded decreasing sequence and so \(y_{n}\) tends to a limit \(y\) say. We write \[\limsup _{n\rightarrow \infty }x_{n}=y.\]

(ii) Let \(a_{n}\) be a sequence of complex numbers. Show that if the sequence \(|a_{n}|^{1/n}\) is unbounded, then \(\sum _{j=0}^{\infty }a_{j}z^{j}\) has radius of convergence \(0\) and if the the sequence \(|a_{n}|^{1/n}\) is bounded and non-zero, then \(\sum _{j=0}^{\infty }a_{j}z^{j}\) has radius of convergence \((\limsup _{n\rightarrow \infty }|a_{n}|^{1/n})^{-1}\). State, with reasons, what happens when \(\limsup _{n\rightarrow \infty }|a_{n}|^{1/n}=0\)

Q 18.10. Suppose that \(A\) and \(B\) are non-empty bounded subsets of \(\mathbb R\). Show that \[\sup \{a+b:a\in A,\ b\in B\}=\sup A+\sup B.\] The last formula is more frequently written \[\sup _{a\in A,\ b\in B}a+b=\sup _{a\in A}a+\sup _{b\in B}b.\]

Suppose, further that \(a_{n}\) and \(b_{n}\) are bounded sequences of real numbers. For each of the following statements either give a proof that it is always true or an example to show that it is sometimes false.

(i) \(\sup _{n}(a_{n}+b_{n})=\sup _{n}a_{n}+\sup _{n}b_{n}\).

(ii) \(\sup _{a\in A,\ b\in B}ab=(\sup _{a\in A}a)(\sup _{b\in B}b)\).

(iii) \(\inf _{a\in A,\ b\in B}a+b=\inf _{a\in A}a+\inf _{b\in B}b\).

Q 18.11. Prove all the statements made in Lemma 4.3, viz:

(i) The limit is unique. That is, if \(\mathbf {a}_{n}\rightarrow \mathbf {a}\) and \(\mathbf {a}_{n}\rightarrow \mathbf {b}\) as \(n\rightarrow \infty \) then \(\mathbf {a}=\mathbf {b}\).

(ii) If \(\mathbf {a}_{n}\rightarrow \mathbf {a}\) as \(n\rightarrow \infty \) and \(n(1)<n(2)<n(3)\ldots \) then \(\mathbf {a}_{n(j)}\rightarrow \mathbf {a}\) as \(j\rightarrow \infty \).

(iii) If \(\mathbf {a}_{n}=\mathbf {c}\) for all \(n\) then \(\mathbf {a}_{n}\rightarrow \mathbf {c}\) as \(n\rightarrow \infty \).

(iv) If \(\mathbf {a}_{n}\rightarrow \mathbf {a}\) and \(\mathbf {b}_{n}\rightarrow \mathbf {b}\) as \(n\rightarrow \infty \) then \(\mathbf {a}_{n}+\mathbf {b}_{n} \rightarrow \mathbf {a}+\mathbf {b}\).

(v) Suppose \(\mathbf {a}_{n}\in {\mathbb R}^{m}\), \(\mathbf {a}\in {\mathbb R}^{m}\) \(\lambda _{n}\in {\mathbb R}\), and \(\lambda \in {\mathbb R}\). If \(\mathbf {a}_{n}\rightarrow \mathbf {a}\) and \(\lambda _{n}\rightarrow \lambda \) then \(\lambda _{n}\mathbf {a}_{n}\rightarrow \lambda \mathbf {a}\).

Q 18.12. Use the Bolzano–Weierstrass theorem to prove the general principle of convergence in \({\mathbb R}^{m}\) (Theorem 4.6).

Q 18.13. By comparing \[\frac {a}{b}\ \text {with}\ \frac {3a+4b}{2a+3b}\] show that given any \(x\in {\mathbb Q}\) with \(x^{2}\leq 2\) we can find a \(y\in {\mathbb Q}\) with \(y>x\) and \(y^{2}\leq 2\). Deduce that the set \(\{x\in {\mathbb Q}:x^{2}<2\}\) has no supremum in \(\mathbb Q\). (The idea of comparing \(a/b\) with \((3a+4b)/(2a+3b)\) goes back as far as Euclid.)

Q 18.14. If \(a<c<b\) and \(E\) is a subset of \(\mathbb R\) explain why if \(E\cap [a,c]\) and \(E\cap [c,b]\) are finite then so is \(E\cap [a,b]\). Use this to give a proof by bisection of Bolzano’s theorem.

Q 18.15. Do Exercise 3.14 showing the uncountability of the reals.

Let \(y_{1}\), \(y_{2}\), …be any sequence of points in \(\mathbb R\). Let \(x_{0}=0\), \(\delta _{0}=1\).

(i) Show that you can construct inductively a sequence of real numbers \(x_{1}\), \(x_{2}\), …and positive numbers \(\delta _{j}\) such that

  (a) \(|x_{n}-x_{n-1}|<\delta _{n-1}/4\),

  (b) \(x_{n}\neq y_{n}\),

  (c) \(0<\delta _{n}<|x_{n}-y_{n}|\),

  (d) \(\delta _{n}<\delta _{n-1}/4\).

(ii) Show that \(\delta _{n+m}<4^{-m}\delta _{n}\) for \(m,n\geq 0\) and deduce that the \(x_{n}\) form a Cauchy sequence. Conclude that \(x_{n}\) tends to a limit \(x\).

(iii) Show that \(|x_{n+m}-x_{n}|<\delta _{n}/3\) for all \(m,n\geq 0\). Deduce that \(|x-x_{n}|\leq \delta _{n}/3\) for all \(n\geq 0\). Why does this show that \(y_{n}\neq x\) for all \(n\).

(iv) Prove that the real numbers are uncountable.

Q 18.16. Suppose we consider the collection \(\mathcal Q\) of rational functions \[f(X)=\frac {a_{0}+a_{1}X+\dots +a_{n}X^{n}} {b_{0}+b_{1}X+\dots +b_{m}X^{m}}\] (where the \(a_{j}\) and \(b_{j}\) are real and where \(b_{m}\neq 0\) and \(a_{n}\neq 0\) if \(n\geq 1\)) with the usual rules for addition and multiplication. Suppose that we say that \(f(X)>0\) if \(a_{n}b_{m}>0\) and that \(f>g\) if \(f-g>0\). Convince yourself that \(\mathcal Q\) has all the standard properties of arithmetic and order we expect. (Some students will only be able to convince themselves by looking up the axioms for an ordered field in, for example, [8] and then verifying each axiom in turn. Good luck to them, I say.)

Show that \(1/n>1/X>0\) for all integers \(n\) and so Archimedes’ Axiom is false for \(\mathcal Q\).

Q 19.1. Here is an alternative proof of Theorem 6.8

(i) Let \(K\geq 0\) and \(\epsilon >0\). If \(f:[a,b]\rightarrow {\mathbb R}\) has the property that \[f(b)-f(a)\geq (K+\epsilon )(b-a)\] and \(N\) is strictly positive integer show that there exists an integer \(r\) with \(1\leq r\leq N\) and \[f(a+r(b-a)/N)-f(a+(r-1)(b-a)/N)\geq (K+\epsilon )(b-a)/N\]

(ii) Under the conditions of part (i) deduce that there exists a sequence of closed intervals \([x_{n},y_{n}]\) with \[[a,b]=[x_{0},y_{0}]\supseteq [x_{1},y_{1}]\supseteq [x_{2},y_{2}]\supseteq \dots \] such that \(f(y_{n})-f(x_{n})\geq (K+\epsilon )(y_{n}-x_{n})\) and \(y_{n}-x_{n}\rightarrow 0\) as \(n\rightarrow \infty \).

(iii) Continuing with the same notation, show that \(x_{n}\rightarrow c\) for some \(c\in [a,b]\). Show that \(c\in [x_{n},y_{n}]\) for all \(n\) and that, if \(c\neq x_{n},y_{n}\) then \[\max \left ( \frac {f(y_{n})-f(c)}{y_{n}-c}, \frac {f(c)-f(x_{n})}{c-x_{n}} \right )\geq K+\epsilon .\] Conclude that if \(f\) is differentiable at \(c\) then \(f'(c)\geq K+\epsilon \).

(iv) Deduce all the results of Theorem 6.8.

Q 19.2. (i) If \(E\) is a set show that the set \[E^{\circ }=\bigcup \{U:\text {$U$ is open and $U\subseteq E$}\}\] has the following properties.

  (a) \(E^{\circ }\) is open.

  (b) \(E^{\circ }\subseteq E\).

  (c) If \(A\) is open and \(A\subseteq E\) then \(A\subseteq E^{\circ }\).

(ii) Suppose that \(E'\) is such that

  (a) \(E'\) is open.

  (b) \(E'\subseteq E\).

  (c) If \(A\) is open and \(A\subseteq E\) then \(A\subseteq E'\).

Show that \(E'=E^{\circ }\).

(iii) Explain why (i) and (ii) make it possible to define the interior of \(E\) as the largest open set contained in \(E\).

(iv) Show why it is possible to define the closure of \(E\) as the smallest closed set containing \(E\).

(v) Find the interior and closure of \((a,b)\), \([a,b)\), \((a,b]\) and \([a,b]\) where \(a<b\).

Q 19.3. Do the Exercise 5.13 restated below.

After looking at parts (iii) to (v) of Lemma 4.3 state the corresponding results for continuous functions. (Thus part (v) corresponds to the statement that if \(\lambda :E\rightarrow {\mathbb R}\) and \(f:E\rightarrow {\mathbb R}^{p}\) are continuous at \({\mathbf x}\in E\) then so is \(\lambda f\).) Prove your statements directly from Definition 5.12.

Q 19.4. If \(f,g:{\mathbb R}^{p}\rightarrow {\mathbb R}^{m}\) are continuous show that \[\begin {pmatrix} f\\g \end {pmatrix} :{\mathbb R}^{p}\rightarrow {\mathbb R}^{2m} ={\mathbb R}^{m}\times {\mathbb R}^{m}\] defined by \[\begin {pmatrix} f\\g \end {pmatrix}({\mathbf x})= \begin {pmatrix} f({\mathbf x})\\g({\mathbf x}) \end {pmatrix}\] is continuous.

Show that the map \(A:{\mathbb R}^{m}\times {\mathbb R}^{m} \rightarrow {\mathbb R}^{m}\) defined by \[A\begin {pmatrix} \mathbf {x}\\ \mathbf {y} \end {pmatrix}=\mathbf {x}+\mathbf {y}\] whenever \(\mathbf {x},\mathbf {y}\in {\mathbb R}^{m}\) is continuous. By applying the continuity of compositions of continuous functions (Lemma 5.16) to \(\displaystyle A\circ \begin {pmatrix} f\\g \end {pmatrix}\) deduce that \(f+g\) is continuous.

Obtain the other results of Question 19.3 similarly.

Q 19.5. Let \(f:{\mathbb R}\rightarrow {\mathbb R}\). State which of the following statements are always true and which may be false giving a proof or a counter example as appropriate.

(i) If \(f^{-1}(E)\) is closed whenever \(E\) is closed then \(f\) is continuous.

(ii) If \(f\) is continuous then \(f(U)\) is open whenever \(U\) is open.

(iii) If \(f\) is continuous then \(f(E)\) is bounded whenever \(E\) is bounded.

(iv) If \(f(E)\) is bounded whenever \(E\) is bounded then \(f\) is continuous.

Q 19.6. Suppose that \(E\) is a subset of \({\mathbb R}^{n}\). Show that \(E\) is closed and bounded if and only if every continuous function \(f:E\rightarrow {\mathbb R}^{n}\) is bounded.

Q 19.7. If \(A\) and \(B\) are subsets of \({\mathbb R}^{m}\) we write \[A+B=\{{\mathbf a}+{\mathbf b}: {\mathbf a}\in A,\ {\mathbf b}\in B\}\]

By considering the case \(m=1\), \[A=\{-n:n\in {\mathbb N}\}\ \text {and} \ B=\{n+1/n:n\in {\mathbb N},\ n\geq 2\}\] show that if \(A\) and \(B\) are closed it does not follow that \(A+B\) is closed.

Show however that if \(A\) is closed and bounded and \(B\) is closed then \(A+B\) is closed.

If \(A\) and \(B\) are open, does it follow that \(A+B\) is open? Give reasons.

Q 19.8. Consider the map \(\Omega :{\mathbb R}^{6}\rightarrow {\mathbb R}^{3}\) given by \[\Omega \begin {pmatrix}{\mathbf x}\\{\mathbf y} \end {pmatrix}= {\mathbf x}\wedge {\mathbf y}\] for all \({\mathbf x},{\mathbf y}\in {\mathbb R}^{3}\). (Here \(\wedge \) is the ‘vector product’ of mathematical methods.) Show directly from the definition that \(\Omega \) is differentiable.

Having done this, find the Jacobian matrix of \(\Omega \).

Q 19.9 (Very traditional). Consider the map \(f:{\mathbb R}^{2}\rightarrow {\mathbb R}\) given by \[f(x,y)=xy|x-y|.\] At which points is \(f\) differentiable? Prove your statements.

[Deal quickly with the easy part. Do not be surprised if the hard part reveals a special case that must be dealt with separately.]

Q 19.10 (Also very traditional). (i) Consider the function \(f:{\mathbb R}\rightarrow {\mathbb R}\) given by \(f(t)=t^{2}\sin 1/t\) for \(t\neq 0\), \(f(0)=0\). Show that \(f\) is everywhere differentiable and find its derivative. Show that \(f'\) is not continuous. [Deal quickly with the easy part and then go back to the definition to deal with \(t=0\). There are wide selection of counter-examples obtained by looking at \(t^{\beta }\sin t^{\alpha }\) for various values of \(\alpha \) and \(\beta \).]

(ii) Find an infinitely differentiable function \(g:(1,\infty )\rightarrow {\mathbb R}\) such that \(g(t)\rightarrow 0\) but \(g'(t)\nrightarrow 0\) as \(t\rightarrow \infty \).

(iii) Consider the function \(F:{\mathbb R}^{2}\rightarrow {\mathbb R}\) given by \(F(s,t)=f(s)+f(t)\) where \(f\) is the function defined in (i). Show that \(F\) is everywhere differentiable but \(F_{,1}\) and \(F_{,2}\) are not continuous at \((0,0)\).

Q 19.11. Let \(\mathbf {u}\) and \(\mathbf {v}\) be linearly independent vectors in \({\mathbb R}^{2}\). Suppose that \(F:{\mathbb R}^{2}\rightarrow {\mathbb R}\) has the property that \[\frac {F({\mathbf x}+h{\mathbf u})-F({\mathbf x})}{h} \rightarrow a({\mathbf x}),\ \text {and} \ \frac {F({\mathbf x}+h{\mathbf v})-F({\mathbf x})}{h} \rightarrow b({\mathbf x})\] as \(h\rightarrow 0\) for all \({\mathbf x}\in {\mathbb R}^{2}\). Suppose further that \(a:{\mathbb R}^{2}\rightarrow {\mathbb R}\) is continuous. Show that \(F\) is everywhere differentiable.

Q 19.12. Show that Definition 7.4 can be modified to give the definition of an operator norm \(\|\alpha \|\) for a linear map \(\alpha :{\mathbb C}^{m}\rightarrow {\mathbb C}^{p}\) and that the results of Lemma 7.6 continue hold mutatis mutandis¹¹.

If \(\alpha \) has matrix \((a_{ij})\) with respect to the standard bases show that \[\max _{i,j}|a_{ij}|\leq \|\alpha \|\leq mp\max _{i,j}|a_{ij}|.\]

For the rest of this question we take \(m=p\) (so we are dealing with endomorphisms of the vector space \({\mathbb C}^{m}\)).

(i) If \(\alpha \) has a lower triangular matrix \((a_{ij})\) with respect to the standard basis show that the set of its eigenvalues is precisely the set of the diagonal entries \(a_{ii}\). (Pure algebra.)

(ii) If \(\alpha \) has a lower triangular matrix \((a_{ij})\) and \(\epsilon >0\) show that we can find \(\beta \) such that \(\beta \) has \(m\) distinct non-zero eigenvalues and \(\|\beta -\alpha \|<\epsilon \).

(iii) In algebra you show that given any endomorphism \(\alpha \) we can find an invertible endomorphism \(\theta \) such that \(\theta \alpha \theta ^{-1}\) has a lower triangular matrix with respect to the standard basis. (N.B. This is true for complex vector spaces but not for real ones.) Use this fact to show that given any endomorphism \(\alpha \) and any \(\eta >0\) we can an endomorphism \(\gamma \) such that \(\|\gamma -\alpha \|<\eta \) and \(\gamma \) has \(m\) distinct non-zero eigenvalues.

(iv) Write \(P_{\alpha }(t)=\det (t\iota -\alpha )\) (where \(\iota \) is the identity endomorphism). Show that if \(\alpha \) has \(m\) distinct eigenvalues then \(P_{\alpha }(\alpha )=0\). (Pure algebra.)

(v) Let \(\alpha \) be any endomorphism. By considering a sequence \(\alpha _{n}\) of endomorphisms with \(m\) distinct eigenvalues such that \(\|\alpha _{n}-\alpha \|\rightarrow 0\) as \(n\rightarrow \infty \) show that \(P_{\alpha }(\alpha )=0\).

[This procedure is substantially longer than that used to prove the Cayley-Hamilton theorem in your algebra course but it is an instructive way of looking at things.]

Q 20.1. At the beginning of the 19th century it was hoped that the calculus could be reduced to mere algebra by using Taylor’s theorem. The following example due to Cauchy shows why such hopes were misplaced.

Let \(E:{\mathbb R}\rightarrow {\mathbb R}\) be defined by \(E(t)=\exp (-1/t^{2})\) for \(t\neq 0\) and \(E(0)=0\). Use induction to show that \(E\) is infinitely differentiable with \begin {align*} E^{(n)}(t)&=Q_{n}(1/t)E(t)\qquad \text {for $t\neq 0$}\\ E^{(n)}(0)&=0. \end {align*}

where \(Q_{n}\) is a polynomial. For which values of \(t\) is it true that \[E(t)=\sum _{n=0}^{\infty }\frac {E^{n}(0)t^{n}}{n!}?\]

Explain why the behaviour of \(E\) is consistent with Lemma 8.7 (taking \(m=1\)).

Q 20.2. In the first year you saw how the find stationary points for a well behaved function \(f:{\mathbb R}^{2}\rightarrow {\mathbb R}\) and how to decide (except in certain specified cases) whether they are maxima, minima or saddle-points. Use Theorem 8.3 to state and prove your results rigorously.

In the course on quadratic mathematics you will see that given any real symmetric \(m\times m\) matrix \(A\) you can find a special orthogonal matrix \(P\) such that \(PAP^{T}\) is diagonal. Explain the relevance of this result to the problem of deciding (except in certain specified cases) when a stationary point of a well behaved function \(f:{\mathbb R}^{m}\rightarrow {\mathbb R}\) is a maximum, a minimum or a saddle point.

Q 20.3. Define \(f:[0,1]\rightarrow {\mathbb R}\) by \(f(p/q)=1/q\) when \(p\) and \(q\) are coprime integers with \(1\leq p <q\) and \(f(x)=0\) otherwise.

(i) Show that \(f\) is Riemann integrable and find \(\int _{0}^{1}f(x)\,dx\).

(ii) At which points is \(f\) continuous? Prove your answer.

Q 20.4. We say that a function \(f:[0,1]\rightarrow {\mathbb R}\) is of bounded variation if there exists a constant \(K\) such that whenever \(0\leq x_{0}\leq x_{1}\leq x_{2}\dots \leq x_{n}\leq 1\) we have \[\sum _{j=1}^{n}|f(x_{j-1})-f(x_{j})|\leq K.\]

(i) Show that if \(g:[0,1]\rightarrow {\mathbb R}\) is increasing then \(g\) is of bounded variation. Show that if \(h_{1},h_{2}:[0,1]\rightarrow {\mathbb R}\) are increasing then \(h=h_{1}-h_{2}\) is of bounded variation.

(ii) Suppose that \(f\) is of bounded variation. Show that \[f_{1}(x)=\sup \{ {\textstyle \sum _{j=1}^{n}f(x_{j})-f(y_{j})} :0\leq y_{1}\leq x_{1}\leq y_{2}\leq x_{2}\leq \dots \leq y_{n}\leq x_{n}\leq x\}\] is a well defined increasing function. Show that \(f_{1}-f\) is also an increasing function and so \(f\) is the difference of two increasing functions.

(iii) Show that if we set \(f(x)=x\sin x^{-1}\) for \(x\neq 0\) and \(f(0)=0\) then \(f:[0,1]\rightarrow {\mathbb R}\) is a continuous function which is not of bounded variation.

[Remember diagrams may not be rigorous but they certainly help you understand what you are talking about.]

Q 20.5. If \(f:[0,1]\rightarrow {\mathbb R}\) is increasing show that if \(x_{n}\) is an increasing sequence then \(f(x_{n})\) tends to limit as \(n\rightarrow \infty \). Deduce carefully that if \(0<y\leq 1\) then \(f(x)\) tends to a limit \(f(y-)\) as \(x\rightarrow y\) through values of \(x\) with \(x<y\). (We shall set \(f(0-)=f(0)\).) Define \(f(y+)\) similarly.

Let us call \(f(y+)-f(y-)\) the jump \(J(y)\) at \(y\). Show that if \(y_{1}\), \(y_{2}\), …\(y_{n}\) are distinct points of \((0,1)\) then \[\sum _{j=1}^{n}J(y_{j})\leq f(1)-f(0).\] Show that \(f\) can have only finitely many jumps of size \(1/n\) or more. Deduce that \(f\) has only countably many jumps and that \(f\) must be continuous at some point.

Q 20.6. Let \(\mathcal K\) be the set of functions \(f:[0,1]\rightarrow {\mathbb R}\) such that \(f\) is continuous on \((0,1]\) and \(f\) has an improper Riemann integral \[{\mathcal I}f=\int _{0}^{1}f(x)\,dx.\]

Show that if \(f,g\in \mathcal K\) and \(\lambda ,\mu \in {\mathbb R}\) then \(\lambda f+\mu g\in \mathcal K\) and \[{\mathcal I}(\lambda f+\mu g)=\lambda {\mathcal I}f +\mu {\mathcal I}g.\]

By giving a proof or counter-example establish whether it is always true that if \(f,g\in {\mathcal K}\) then \(fg\in {\mathcal K}\).

Q 20.7. Suppose that \(f:{\mathbb R}^{2}\rightarrow {\mathbb R}\) is continuous. By using results on the differentiation of integrals which should be quoted exactly show that \[\frac {d\ }{dt}\int _{a}^{t} \left (\int _{c}^{d}f(u,v)\,dv\right )\,du =\frac {d\ }{dt}\int _{c}^{d} \left (\int _{a}^{t}f(u,v)\,du\right )\,dv.\] Deduce that \[\int _{a}^{b} \left (\int _{c}^{d}f(u,v)\,dv\right )\,du =\int _{c}^{d} \left (\int _{a}^{b}f(u,v)\,du\right )\,dv,\] or, more informally \[\int _{a}^{b} \int _{c}^{d}f(u,v)\,dv\,du =\int _{c}^{d} \int _{a}^{b}f(u,v)\,du\,dv.\]

Q 20.8. Prove all the results of Section 11. (A long but easy and useful exercise. In order to keep the noise level from whinging students down to tolerable levels this counts as 2 questions.)

Q 20.9. See Question 20.8.

Q 20.10. Consider \(\mathcal E\) the space of linear maps \(\alpha :{\mathbb R}^{u}\rightarrow {\mathbb R}^{u}\). We wish to show that \(d(\alpha ,\beta )=\|\alpha -\beta \|\) (with \(\|\alpha \|\) the usual ‘operator norm’) defines a complete metric.

To this end consider a Cauchy sequence \(\alpha _{n}\) in \(\mathcal E\). Show that if \(\mathbf {x}\in {\mathbb R}^{u}\) then \(\alpha _{n}(\mathbf {x})\) tends to limit, call it \(\alpha (\mathbf {x})\). Now show that \(\alpha :{\mathbb R}^{u}\rightarrow {\mathbb R}^{u}\) is linear. By using the inequality \begin {align*} \|\alpha (\mathbf {x})-\alpha _{n}(\mathbf {x})\|& \leq \|\alpha _{m}(\mathbf {x})-\alpha _{n}(\mathbf {x})\| +\|\alpha _{m}(\mathbf {x})-\alpha (\mathbf {x})\|\\ &\leq \sup _{p,q\geq n}\|\alpha _{p}(\mathbf {x})-\alpha _{q}(\mathbf {x})\| +\|\alpha _{m}(\mathbf {x})-\alpha (\mathbf {x})\|, \end {align*}

and allowing \(m\rightarrow \infty \) show that \[\|\alpha -\alpha _{n}\|\leq \sup _{p,q\geq n}\|\alpha _{p}-\alpha _{q}\|,\] and deduce that \(\|\alpha -\alpha _{n}\|\rightarrow \infty \) as \(n\rightarrow \infty \).

Q 20.11. In Question 20.10 we showed that \(\mathcal {E}\) with the appropriate metric is complete. In this question we make use of the result.

(i) Suppose \(\alpha \in \mathcal {E}\) and \(\|\alpha \|<1\). Let \[\sigma _{n}=\iota +\alpha +\alpha ^{2}+\dots +\alpha ^{n}.\] Show that \(\sigma _{n}\) converges to a limit \(\sigma \), say.

Compute \((\iota -\alpha )\sigma _{n}\) and show, carefully that \((\iota -\alpha )\sigma =\iota \).

(iii) Deduce that if \(\|\iota -\gamma \|<1\) then \(\gamma \) is invertible.

(iv) Show that if \(\beta \in \mathcal {E}\) is invertible and \(\|\beta -\tau \|<\|\beta ^{-1}\|^{-1}\) then \(\tau \) is invertible.

(v) Conclude that the invertible elements of \(\mathcal E\) form an open set.

Q 20.12. Consider \(C([-1,1])\) the set of continuous functions \(f:[-1,1]\rightarrow {\mathbb R}\). Show that, if we set \[d(f,g)=\|f-g\|_{2}= \left (\int _{-1}^{1}|f(x)-g(x)|^{2}\,dx\right )^{1/2}\] then \(d\) is a metric but \(d\) is not complete.

Show that if \(\|f-g\|_{1}\) is defined as in Example 13.8 then \(2^{1/2}\|f\|_{2}\geq \|f\|_{1}\) for all \(f\in C([-1,1])\) but that, given any constant \(K>0\) we can find a \(g\in C([-1,1])\) such that \(\|g\|_{2}\geq K\|g\|_{1}\).

Q 20.13. We use the notation of Questions 20.10 and 20.11.

(i) Show that if \(\alpha \in \mathcal {E}\) we can find \(e^{\alpha }\in \mathcal {E}\) such that \[\left \|\sum _{r=0}^{n}\frac {\alpha ^{r}}{r!} -e^{\alpha }\right \|\rightarrow 0\] as \(n\rightarrow \infty \).

(ii) Show carefully that if \(\alpha \) and \(\beta \) commute \[e^{\alpha }e^{\beta }=e^{\alpha +\beta }.\]

(iii) Show that if \(\alpha \) and \(\beta \) are general (not necessarily commuting) elements of \(\mathcal E\) then \[\left \|h^{-2}(e^{h\alpha }e^{h\beta }-e^{h\beta }e^{h\alpha }) -(\alpha \beta -\beta \alpha )\right \|\rightarrow 0\] as the real number \(h\rightarrow 0\).

Conclude that, in general, \(e^{\alpha }e^{\beta }\) and \(e^{\alpha +\beta }\) need not be equal.

Q 21.1. Consider the two railway metrics \(d_{1}\) and \(d_{2}\) on \({\mathbb R}^{2}\) given in Examples 11.3 and 11.4. For each of the two metrics give a reasonably simple description of the open balls of radius \(r\) and centre \(\mathbf x\) when \({\mathbf x}={\mathbf 0}\) and when \({\mathbf x}\neq {\mathbf 0}\) and (a) \(r\leq \|{\mathbf x}\|\), (b) \(r<\|{\mathbf x}\|\leq 2r\), (c) \(2r<\|{\mathbf x}\|\).

Explain the statement ‘when finding open sets only balls of small radius are important so cases (b) and (c) are irrelevant’.

Give the simplest description you can find of open sets in the two metrics.

Q 21.2. Let \(f:{\mathbb R}^{2}\rightarrow {\mathbb R}\) be differentiable and let \(g(x)=f(x,c-x)\) where \(c\) constant. Show that \(g:{\mathbb R}\rightarrow {\mathbb R}\) is differentiable and find its derivative

(i) directly from the definition of differentiability

and also

(ii) by using the chain rule.

Deduce that if \(f_{,1}=f_{,2}\) throughout \(\mathbb R\) then \(f(x,y)=h(x+y)\) for some differentiable function \(h\).

Q 21.3 (Traditional). Consider the functions \(f_{n}:[0,1]\rightarrow {\mathbb R}\) defined by \(f_{n}(x)=n^{p}x\exp (-n^{q}x)\) where \(p,q>0\).

(i) Show that \(f_{n}\) converges pointwise on \([0,1]\).

(ii) Show that if \(p<q\) then \(f_{n}\) converges uniformly on \([0,1]\).

(iii) Show that if \(p\geq q\) then \(f_{n}\) does not converge uniformly on \([0,1]\). Does \(f_{n}\) converge uniformly on \([0,1-\epsilon ]\)? Does \(f_{n}\) converge uniformly on \([\epsilon ,1]\)? (Here \(0<\epsilon <1\), you should justify your answers.)

Q 21.4. Let \((X,d)\) be a metric space. We say that \(E\) is dense in \(X\) if whenever \(x\in X\) we can find \(e_{n}\in E\) with \(e_{n}\rightarrow x\).

(i) Show that the rationals are dense in the reals (i.e. \(\mathbb Q\) is dense in \(\mathbb R\) if we give \(\mathbb R\) the usual metric). Show that the irrationals are dense in the reals.

[You should make explicit any use you make of the axiom of Archimedes.]

(ii) Consider the space \(C([0,1])\) of continuous functions with the uniform norm. Show that the piecewise linear functions are dense in \(C([0,1])\).

Q 21.5. Recall from your courses in mathematical methods that, if \(f:{\mathbb R}\rightarrow {\mathbb C}\) is a continuous periodic function with period \(2\pi \) then we write \[\hat {f}(n)=\frac {1}{2\pi } \int _{-\pi }^{\pi }f(t)\exp (-int)\,dt.\]

Show, quoting carefully any results that you use, that, if \(a_{n}\in {\mathbb C}\) and \(\sum _{n=-N}^{N} |a_{n}|\) converges, then \(\sum _{n=-N}^{N} a_{n}\exp (int)\) converges uniformly to a continuous periodic function \(f(t)\), say, with period \(2\pi \). Show that \(\hat {f}(n)=a_{n}\). for each \(n\).

[This chain of thought is continued in Question 21.14.]

Q 21.6. Recall that \(\|\ \|\) is said to be a norm on a real vector space \(V\) if the following conditions hold.

(1) \(\|\mathbf {x}\|\geq 0\) for all \(\mathbf {x}\in V\).

(2) \(\|\mathbf {x}\|=0\) implies \(\mathbf {x}=\mathbf {0}\).

(3) \(\|\lambda \mathbf {x}\|=|\lambda |\|\mathbf {x}\|\) for all \(\lambda \in {\mathbb R}\) and all \(\mathbf {x}\in V\).

(4) \(\|\mathbf {x}+\mathbf {y}\|\leq \|\mathbf {x}\|+ \|\mathbf {y}\|\) for all \(\mathbf {x},\mathbf {y}\in {\mathbb R}\).

(i) Let \(\|\ \|_{2}\) be the usual Euclidean norm on \({\mathbb R}^{m}\) and let \(\|\ \|\) be another norm on \({\mathbb R}^{m}\). Let \(I:({\mathbb R}^{m},\|\ \|_{2})\rightarrow ({\mathbb R}^{m},\|\ \|)\) be the identity map, given by \(I\mathbf {x}=\mathbf {x}\). By considering basis vectors or otherwise that there exists a constant \(K\) such that \(\|\mathbf {x}\|\leq K\|\mathbf {x}\|_{2}\) for all \(\|\mathbf {x}\|\in {\mathbb R}^{m}\). Deduce that \(I\) is continuous.

By applying a theorem on continuous functions on closed bounded sets show that there exists a \(k>0\) such that \(\|\mathbf {x}\|\geq k\) for all \(\mathbf {x}\) with \(\|\mathbf {x}\|_{2}=1\). Conclude that \[k\|\mathbf {x}\|_{2}\leq \|\mathbf {x}\|\leq K\|\mathbf {x}\|_{2}\] for all \(\|\mathbf {x}\|\in {\mathbb R}^{m}\).

[Thus all norms on a finite dimensional space are essentially the same.]

(ii) Consider the real vector space \(\mathbb {R}^{\mathbb {N}}\) of all real sequences \(\mathbf {x}=(x_{1},x_{2},\dots )\) with the usual vector addition and multiplication by scalars. Let \(V\) be the set of all sequences \(\mathbf {x}=(x_{1},x_{2},\dots )\) such that \(x_{j}\neq 0\) for only finitely many \(j\). Show that \(V\) is a subspace of \(\mathbb {R}^{\mathbb {N}}\) and so a vector space in its own right. By considering norms of the form \(\|\mathbf {x}\|=\max _{j}\kappa _{j}|x_{j}|\), or otherwise, find two norms \(\|\ \|_{A}\) and \(\|\ \|_{B}\) such that \[\sup _{\|\mathbf {x}\|_{A}=1}\|\mathbf {x}\|_{B} =\sup _{\|\mathbf {x}\|_{B}=1}\|\mathbf {x}\|_{A}=\infty .\]

Q 21.7. (i) Suppose that \(f:{\mathbb R}\rightarrow {\mathbb R}\) is a twice differentiable function such that \[\left |\frac {f(x)f''(x)}{f'(x)^{2}}\right | \leq \lambda \] for all \(x\) and some \(|\lambda |<1\) Show that the mapping \[Tx=x-\frac {f(x)}{f'(x)}\] is a contraction mapping and deduce that \(f\) has a unique root \(y\).

(ii) Suppose that \(F:{\mathbb R}\rightarrow {\mathbb R}\) is a twice differentiable function such that \[\left |\frac {F(x)F''(x)}{F'(x)^{2}}\right | \leq \lambda \] for all \(|x|\leq a\) and some \(|\lambda |<1\) and that \(F(0)=0\). Consider the mapping \[Tx=x-\frac {F(x)}{F'(x)}.\] Show that \(T^{n}x\rightarrow 0\).

Suppose that \[\frac {\sup _{|t|\leq a}|f'(t)|\sup _{|t|\leq a}|f''(t)|} {\inf _{|t|\leq a}|f'(t)|^{2}}=M.\] By using the mean value theorem twice, show that if \(|x|\leq a\) then \[|Tx|\leq Mx^{2}.\]

(iii) If you know what the Newton–Raphson method is, comment on the relevance of the results of (i) and (ii) to that method.

Q 21.8. We work in \({\mathbb R}^{m}\) with the usual ‘Euclidean norm’. The following line of thought is extremely important in later work. Suppose we want the solution \({\mathbf x}_{0}\) of \[{\mathbf x}+{\boldsymbol \epsilon }({\mathbf x})= {\mathbf y}\] where \({\boldsymbol \epsilon }({\mathbf x})\) is ‘a small error term’ or ‘of first order compared to \(\mathbf x\). The following method of solution is traditional (for the excellent reason that it usually works like a Spanish charm). Start by ignoring the small \(\boldsymbol \epsilon \) term and guess \({\mathbf x}_{0}\approx {\mathbf x}_{1}={\mathbf y}\). Since this initial guess is good we estimate the small \(\boldsymbol \epsilon \) term as \({\boldsymbol \epsilon }({\mathbf x}_{1})\). Feeding this estimate back into the equation we now guess \({\mathbf x}_{0}\approx {\mathbf x}_{2}={\mathbf y}-{\boldsymbol \epsilon }({\mathbf x}_{1})\). Repeating this process gives us the rule for successive approximations \[\mathbf {x}_{n+1}=\mathbf {y}-{\boldsymbol \epsilon }(\mathbf {x}_{n}).\] We now try to justify this in certain cases.

(i) Suppose that \({\boldsymbol \epsilon }({\mathbf 0})={\mathbf 0}\) and \[\|{\boldsymbol \epsilon }({\mathbf a})- {\boldsymbol \epsilon }({\mathbf b})\| <\|{\mathbf a}-{\mathbf b}\|/2\] for all \(\|\mathbf {a}\|,\|\mathbf {b}\|\leq \delta \) where \(\delta >0\). (Note that these are rather stronger conditions than might have been expected. However, this is not to say that the general idea might not work under weaker conditions.) We shall show that the method works for \(\|{\mathbf y}\|\leq \delta /2\).

Consider the closed ball \(B=\{{\mathbf x}:\|{\mathbf x}\|\leq \delta \}\). Show that the equation \[T({\mathbf x})=\mathbf {y}-{\boldsymbol \epsilon }({\mathbf x})\] defines a map \(T:B\rightarrow B\). Now use the contraction mapping theorem to show that \[{\mathbf x}+{\boldsymbol \epsilon }({\mathbf x})= {\mathbf y}\] has a unique solution \({\mathbf x}(\mathbf {y})\) with \(\|{\mathbf x}(\mathbf {y})\|\leq \delta \) and that \(T^{n}({\mathbf y})\rightarrow {\mathbf x}(\mathbf {y})\) as \(n\rightarrow \infty \).

(ii) Suppose that \(f:{\mathbb R}^{m}\rightarrow {\mathbb R}^{m}\) is a differentiable function with derivative continuous at \(\mathbf 0\), \(f({\mathbf 0})={\mathbf 0}\) and \(Df({\mathbf 0})=I\). Show that we can find a \(\delta >0\) such that whenever \(\|{\mathbf y}\|\leq \delta /2\) the equation \[f({\mathbf x})={\mathbf y}\] has a unique solution \({\mathbf x}(\mathbf {y})\) with \(\|{\mathbf x}(\mathbf {y})\|\leq \delta \).

(iii) Suppose that \(f:{\mathbb R}^{m}\rightarrow {\mathbb R}^{m}\) is a differentiable function with derivative continuous at \(\mathbf 0\), \(f({\mathbf 0})={\mathbf 0}\) and \(Df({\mathbf 0})\) invertible. Show that we can find a \(\delta _{1},\delta _{2}>0\) such that whenever \(\|{\mathbf y}\|\leq \delta _{1}\) the equation \[f({\mathbf x})={\mathbf y}\] has a unique solution \({\mathbf x}(\mathbf {y})\) with \(\|{\mathbf x}(\mathbf {y})\|\leq \delta _{2}\).

Q 21.9. (A method of Abel) (i) Suppose that \(a_{j}\) and \(b_{j}\) are sequences of complex numbers and that \(S_{n}=\sum _{j=1}^{n}a_{j}\) for \(n\geq 1\) and \(S_{0}=0\). Show that, if \(1\leq u\leq v\) then \[\sum _{j=u}^{v}a_{j}b_{j}=\sum _{j=u}^{v}S_{j}(b_{j}-b_{j+1}) -S_{u-1}b_{u}+S_{v}b_{v+1}.\] (This is known as partial summation, for obvious reasons.)

(ii) Suppose now that, in addition, the \(b_{j}\) form a decreasing sequence of positive terms and that \(|S_{n}|\leq K\) for all \(n\). Show that \[\left |\sum _{j=u}^{v}a_{j}b_{j}\right | \leq 2Kb_{u}.\] Deduce that if \(b_{j}\rightarrow 0\) as \(j\rightarrow \infty \) then \(\sum _{j=1}^{\infty }a_{j}b_{j}\) converges.

Deduce the alternating series test.

(iii) If \(b_{j}\) is a decreasing sequence of positive terms with \(b_{j}\rightarrow 0\) as \(j\rightarrow \infty \) show that \(\sum _{j=1}^{\infty }b_{j}z^{j}\) converges uniformly in the region given by \(|z|\leq 1\) and \(|z-1|\geq \epsilon \) for all \(\epsilon >0\).

Q 21.10. We work in \({\mathbb R}^{m}\) with the usual distance.

(i) Show that if \(A_{1}\), \(A_{2}\), are non-empty, closed and bounded with \(A_{1}\supseteq A_{2}\supseteq \dots \) then \(\bigcap _{j=1}^{\infty }A_{j}\) is non empty. Is this result true if we merely assume \(A_{j}\) closed and non-empty? Give reasons.

(ii) If \(A\) is non-empty, closed and bounded show that we can find \({\mathbf a}',{\mathbf b}'\in A\) such that \[\|{\mathbf a}'-{\mathbf b}'\|\geq \|{\mathbf a}-{\mathbf b}\|\] for all \({\mathbf a},{\mathbf b}\in A\). Is this result true if we merely assume \(A\) bounded and non-empty? Give reasons.

Q 21.11. We work in \({\mathbb R}^{m}\) with the usual distance. Let \(E\) be a closed non-empty subset of \({\mathbb R}^{m}\) and let \(T\) be a map \(T:E\rightarrow E\).

(i) Suppose \(\|T({\mathbf a})-T({\mathbf b})\|<\|{\mathbf a}-{\mathbf b}\|\) for all \({\mathbf a},{\mathbf b}\in E\) with \({\mathbf a}\neq {\mathbf b}\). We saw in Example 15.2 that \(T\) need not have a fixed point. Show that, if \(T\) has a fixed point, it is unique.

(ii) Suppose \(\|T({\mathbf a})-T({\mathbf b})\|>\|{\mathbf a}-{\mathbf b}\|\) for all \({\mathbf a},{\mathbf b}\in E\) with \({\mathbf a}\neq {\mathbf b}\). In Question 21.17 it is shown that \(T\) need not have a fixed point. Show that, if \(T\) has a fixed point, it is unique.

(iii) Suppose \(\|T({\mathbf a})-T({\mathbf b})\|=\|{\mathbf a}-{\mathbf b}\|\) for all \({\mathbf a},{\mathbf b}\in E\). Show that \(T\) need not have a fixed point. and that, if \(T\) has a fixed point, it need not be unique.

(iv) Suppose now that \(E\) is non-empty, closed and bounded and \[\|T({\mathbf a})-T({\mathbf b})\|<\|{\mathbf a}-{\mathbf b}\|\] for all \({\mathbf a},{\mathbf b}\in E\) with \({\mathbf a}\neq {\mathbf b}\). By considering \(\inf _{{mathbf x}\in E}\|{\mathbf x}-T({mathbf x})\|\), or otherwise show that \(T\) has a fixed point.

Q 21.12. From time to time numerical analysts mention the spectral radius. It forms no part of any of non-optional 1B courses but the reader may be interested to see what it is.

(i) Give an example of a linear map \(\beta :{\mathbb R}^{m}\rightarrow {\mathbb R}^{m}\) such that \(\beta ^{m-1}\neq {\mathbf 0}\) but \(\beta ^{m}={\mathbf 0}\).

(ii) Let \(\alpha :{\mathbb R}^{m}\rightarrow {\mathbb R}^{m}\) be linear. If \(n=jk+r\) explain why \[\|\alpha ^{jk+r}\|\leq \|\alpha ^{j}\|^{k}\|\alpha \|^{r}.\]

(iii) Continuing with the hypotheses of (ii), show that \(\Delta =\inf _{n}\|\alpha ^{n}\|^{1/n}\) is well defined and, by using the result of (ii), or otherwise, that \[\|\alpha ^{n}\|^{1/n}\rightarrow \Delta .\] We call \(\Delta \) the spectral radius of \(\alpha \) and write \(\rho (\alpha )=\Delta \).

(iv) If \(\alpha :{\mathbb R}^{m}\rightarrow {\mathbb R}^{m}\) is diagonalisable show that \[\rho (\alpha )=\max \{|\lambda |:\lambda \ \text {an eigenvalue of $\alpha $}\}.\]

(v) Give an example of linear maps \(\alpha ,\beta :{\mathbb R}^{2}\rightarrow {\mathbb R}^{2}\) such that \(\rho (\alpha )=\rho (\beta )=0\) but \(\rho (\alpha +\beta )=1\).

(vi) Recalling Question 20.11 of the third sheet, show that if \(\rho (\iota -\gamma )<1\) then \(\gamma \) is invertible.

Q 21.13. We work in \({\mathbb R}^{m}\), \(\|{\mathbf x}-{\mathbf y}\|\) will represent the usual Euclidean distance between \(\mathbf x\) and \(\mathbf y\).

(i) If \(K\) is a closed non-empty bounded set and \(\mathbf x\) is any point, show that there exists a point \({\mathbf k}'\in K\) such that \[\|{\mathbf x}-{\mathbf k}\|\geq \|{\mathbf x}-{\mathbf k}'\|\] for all \({\mathbf k}\in K\). Is \({\mathbf k}'\) necessarily unique?

(ii) If \(E\) is a non-empty closed set and \(\mathbf x\) is any point, show that there exists a point \({\mathbf e}'\in E\) such that \[\|{\mathbf x}-{\mathbf e}\|\geq \|{\mathbf x}-{\mathbf e}'\|\] for all \({\mathbf e}\in E\). We write \(d({\mathbf x},E)=\|{\mathbf x}-{\mathbf e}'\|.\)

(iii) With the notation of (ii) show that \[d({\mathbf x},E)+\|{\mathbf x}-{\mathbf y}\| \geq d({\mathbf y},E).\] Show that \(d(\ ,E):{\mathbb R}^{m}\rightarrow {\mathbb R}\) is continuous.

(iv) If \(E\) is closed and non-empty and \(K\) closed, bounded and non-empty show that there exist \({\mathbf e}'\in E\) and \({\mathbf k}'\in K\) such that \[\|{\mathbf e}-{\mathbf k}\|\geq \|{\mathbf e}'-{\mathbf k}'\|.\] Would this result be true if we only assumed \(E\) and \(K\) closed and non-empty.

Q 21.14. (i) Show that if \(f:{\mathbb R}\rightarrow {\mathbb C}\) is a continuous periodic function with period \(2\pi \) such that \(\hat {f}(n)=0\) for all \(n\) then \[\int _{-\pi }^{\pi }(1-\epsilon _{1}+\epsilon _{2}\cos t)^{N}f(t)\,dt=0\] for all \(N\).

(ii) Show that, given \(\delta >0\) we can find \(\epsilon _{1},\ \epsilon _{2}>0\) and an \(\eta >0\) such that \begin {align*} (1-\epsilon _{1}+\epsilon _{2}\cos t)^{N}\rightarrow \infty &\ \text {uniformly for $|t|\leq \eta $}\\ (1-\epsilon _{1}+\epsilon _{2}\cos t)^{N}\rightarrow 0&\ \text {uniformly for $\delta \leq |t|\leq \pi $} \end {align*}

as \(N\rightarrow \infty \).

(iii) Show that if \(f:{\mathbb R}\rightarrow {\mathbb C}\) is a continuous periodic function with period \(2\pi \) such that \(f(0)\) is real and \(f(0)>0\) we can find \(\epsilon _{1},\ \epsilon _{2}>0\) such that \[\Re \int _{-\pi }^{\pi }(1-\epsilon _{1}+\epsilon _{2}\cos t)^{N}f(t)\,dt \rightarrow \infty \] as \(N\rightarrow \infty \).

(iv) Show that if \(f:{\mathbb R}\rightarrow {\mathbb C}\) is a continuous periodic function with period \(2\pi \) such that \(\hat {f}(n)=0\) for all \(n\) then \(f(t)=0\) for all \(t\).

(v) By using Question 21.5, or otherwise, show that if if \(F:{\mathbb R}\rightarrow {\mathbb C}\) is a continuous periodic function with period \(2\pi \) such that \(\sum |\hat {F}(n)|\) converges then \[F(t)=\sum _{n=-\infty }^{\infty }\hat {F}(n)\exp (int).\]

Q 21.15. (i) Suppose that \(g:{\mathbb R}\rightarrow {\mathbb C}\) is a continuous periodic function with period \(2\pi \). Show that \[\frac {1}{2\pi }\int _{0}^{2\pi } \left |g(t)-\sum _{r=-N}^{N}\hat {g}(r)\exp (irt)\right |^{2} \,dt =\frac {1}{2\pi }\int _{0}^{2\pi }|g(t)|^{2}\,dt -\sum _{r=-N}^{N}|\hat {g}(r)|^{2}.\] Deduce that \[\sum _{r=-N}^{N}|\hat {g}(r)|^{2} \leq \frac {1}{2\pi }\int _{0}^{2\pi }|g(t)|^{2}\,dt.\] (This is a version of Bessel’s inequality.)

(ii) Now suppose that \(f:{\mathbb R}\rightarrow {\mathbb C}\) is a continuously differentiable periodic function with period \(2\pi \). If \(f\) has derivative \(g\), obtain a simple relation between \(\hat {f}(n)\) and \(\hat {g}(n)\). By applying the Cauchy-Schwarz inequality to \(\sum _{n\neq 0,\ |n|\leq N} |n||\hat {g}(n)|\) show that \[\sum _{n\neq 0,\ |n|\leq N} |\hat {f}(n)| \leq A \frac {1}{2\pi }\int _{0}^{2\pi }|g(t)|^{2}\,dt,\] for some constant \(A\). Conclude that \(\sum |\hat {f}(n)|\) converges.

(iii) Deduce, using Question 21.14, that if \(f:{\mathbb R}\rightarrow {\mathbb C}\) is a continuously differentiable periodic function with period \(2\pi \) then \[f(t)=\sum _{n=-\infty }^{\infty }\hat {f}(n)\exp (int).\]

Q 21.16. We continue with the ideas of Question 21.13. From now on all sets will belong to the collection \(\mathcal {K}\) of closed bounded non-empty subsets of \({\mathbb R}^{m}\). Define \[d(E,F)=\sup _{e\in E}d(e,F)+\sup _{f\in F}d(f,E).\] Show that \(d\) is a metric on \(\mathcal {K}\).

Show also that \(d\) is a complete metric.

Q 21.17. If \((X,d)\) is a complete metric space and \(T:X\rightarrow X\) is a surjective map such that \[d(Tx,Ty)\geq Kd(x,y)\] for all \(x,y\in X\) and some \(K>1\) show that \(T\) has a unique fixed point.

By considering the map \(T:{\mathbb R}\rightarrow {\mathbb R}\) defined by \(T(x)=1+4n+2x\) for \(0\leq x<1\) and \(n\) an integer, or otherwise show that the condition \(T\) surjective can not be dropped.

Q 21.18 (A continuous nowhere differentiable function). The following construction is due to Van der Waerden.

(i) Sketch the graph of the function \(g\) given by the condition \[g(x+k)=|x|\ \ \ \ \mbox {if $k$ is any integer and $-1/2<x\leq 1/2$.}\]

(ii) Let \(g_{n}(t)=2^{-n}g(8^{n}t)\). Sketch the graphs of \(g_{1}\), \(g_{2}\) and \(g_{3}\).

(iii) Let \(f_{n}(t)=g_{1}(t)+g_{2}(t)+\ldots +g_{n}(t)\). Sketch the graphs of \(f_{1}\), \(f_{2}\) and \(f_{3}\).

(iv) Show that \(f_{n}\) converges uniformly to a continuous function \(f\).

(v) If \(n\) is a positive integer and \(r\) an odd integer give an expression for \(f(r2^{-n})\). Show that when \(n\) is large then \(2^{n}|f((r+2)2^{-n})-f(r2^{-n})|\) is large. Conclude that if \(r2^{-n}<t<(r+2)2^{-n}\) then \[\max \left (\left |\frac {f(t)-f(r2^{-n})}{t-r2^{-n}}\right |, \left |\frac {f((r+2)2^{-n})-f(t)}{(r+2)2^{-n}-t}\right |\right ).\]

(vi) By developing the ideas of part (v) show that \(f\) is nowhere differentiable.