For any $S⊂ℝ$ containing 0 , let $f:[0,∞)→ℝ$ be $f(x)=\sup (S∩[0, x])$.
(a) State a formula for $f$ if $S=ℤ$, the integers (no proof necessary).
(b) State a formula for $f$ if $S=ℝ$, all real numbers (no proof necessary).
(c) Prove that any such $f$ is integrable on $[0,10]$.
(a) $f(x)=[x]$; (b) $f(x)=x$; (c) If $x≤y$, then $S∩[0, x]⊂S∩[0, y]$, so any upper bound (such as sup $S∩[0, y]$ ) for the latter is an upper bound for the former, so the least upper bound must satisfy $\sup S∩[0, x]≤\sup S∩[0, y]$, so $f(x)≤f(y)$. Hence $f$ is monotonic, hence integrable on $[0,10]$.