4.5.E: Problems on Monotone Functions
Complete the proofs of Theorems 1 and \(2 .\) Give also an independent (analogous) proof for nonincreasing functions.
Discuss Examples \((\mathrm{d})\) and \((\mathrm{e})\) of §1 again using Theorems \(1-3\).
Show that Theorem 3 holds also if \(f\) is piecewise monotone on \((a, b),\) i.e., monotone on each of a sequence of intervals whose union is \((a, b) .\)
Consider the monotone function \(f\) defined in Problems 5 and 6 of Chapter 3, §11. Show that under the standard metric in \(E^{1}, f\) is continuous on \(E^{1}\) and \(f^{-1}\) is continuous on \((0,1) .\) Additionally, discuss continuity under the metric \(\rho^{\prime} .\)
\(\Rightarrow\) 5. Prove that if \(f\) is monotone on \((a, b) \subseteq E^{*},\) it has at most countably many discontinuities in \((a, b)\).
[Hint: Let \(f \uparrow .\) By Theorem \(3,\) all discontinuities of \(f\) correspond to mutually disjoint intervals \(\left(f\left(p^{-}\right), f\left(p^{+}\right)\right) \neq \emptyset .\) (Why?) Pick a rational from each such interval, so these rationals correspond one to one to the discontinuities and form a countable set (Chapter 1, §9)].
Continuing Problem 17 of Chapter 3, §14, let
\[
G_{11}=\left(\frac{1}{3}, \frac{2}{3}\right), \quad G_{21}=\left(\frac{1}{9}, \frac{2}{9}\right), G_{22}=\left(\frac{7}{9}, \frac{8}{9}\right), \text { and so on; }
\]
that is, \(G_{m i}\) is the \(i\) th open interval removed from \([0,1]\) at the \(m\) th step of the process \( (i=1,2, \ldots, 2^{m-1}, m=1,2, \ldots \text { ad infinitum} )\).
Define \(F :[0,1] \rightarrow E^{1}\) as follows:
(i) \(F(0)=0\);
(ii) if \(x \in G_{m i},\) then \(F(x)=\frac{2 i-1}{2^{m}} ;\) and
(iii) if \(x\) is in none of the \(G_{m i}(\text { i.e., } x \in P),\) then
\[
F(x)=\sup \{ F(y) | y \in \bigcup_{m, i} G_{m i}, y<x \} .
\]
Show that \(F\) is nondecreasing and continuous on \([0,1]\). (\(F\) is called Cantor's function.)
Restate Theorem 3 for the case where \(f\) is monotone on \(A,\) where \(A\) is a (not necessarily open) interval. How about the endpoints of \(A ?\)