4.9.E: Problems on the Darboux Property and Related Topics
Prove Note 1.
Prove Note 3.
Prove continuity at 0 in Example \((\mathrm{c})\).
Prove Theorem 1 for polygon-connected sets.
[Hint: If
\[
B \supseteq \bigcup_{i=0}^{m-1} L\left[\overline{p}_{i}, \overline{p}_{i+1}\right]
\]
with
\[
f\left(\overline{p}_{0}\right)<c<f\left(\overline{p}_{m}\right) ,
\]
show that for at least one \(i,\) either \(c=f\left(\overline{p}_{i}\right)\) or \(f\left(\overline{p}_{i}\right)<c<f\left(\overline{p}_{i+1}\right) .\) Then replace \(\left.B \text { in the theorem by the convex segment } L\left[\overline{p}_{i}, \overline{p}_{i+1}\right] .\right]\)
Show that, if \(f\) is strictly increasing on \(B \subseteq E,\) then \(f^{-1}\) has the same property on \(f[B],\) and both are one to one; similarly for decreasing functions.
For functions on \(B=[a, b] \subset E^{1},\) Theorem 1 can be proved thusly: If
\[
f(a)<c<f(b) ,
\]
let
\[
P=\{x \in B | f(x)<c\}
\]
and put \(r=\sup P\).
Show that \(f(r)\) is neither greater nor less than \(c,\) and so necessarily \(f(r)=c .\)
[Hint: If \(f(r)<c\), continuity at \(r\) implies that \(f(x)<c\) on some \(G_{r}(\delta)\) (§2, Problem 7), contrary to \(r=\sup P\). (Why?)]
Continuing Problem \(4,\) prove Theorem 1 in all generality, as follows.
Define
\[
g(t)=\overline{p}+t(\overline{q}-\overline{p}), \quad 0 \leq t \leq 1 .
\]
\(\text { Then } g \text { is continuous (by Theorem } 3 \text { in } §3)\), and so is the composite function \(h=f \circ g,\) on \([0,1] .\) By Problem \(4,\) with \(B=[0,1],\) there is a \(t \in(0,1)\) with \(h(t)=c .\) Put \(\overline{r}=g(t),\) and show that \(f(\overline{r})=c\).
Show that every equation of odd degree, of the form
\[
f(x)=\sum_{k=0}^{n} a_{k} x^{k}=0 \quad\left(n=2 m-1, a_{n} \neq 0\right) ,
\]
has at least one solution for \(x\) in \(E^{1}\).
[Hint: Show that \(f\) takes both negative and positive values as \(x \rightarrow-\infty\) or \(x \rightarrow+\infty\); thus by the Darboux property, \(f\) must also take the intermediate value 0 for some \(\left.x \in E^{1} .\right]\)
Prove that if the functions \(f : A \rightarrow(0,+\infty)\) and \(g : A \rightarrow E^{1}\) are both continuous, so also is the function \(h : A \rightarrow E^{1}\) given by
\[
h(x)=f(x)^{g(x)} .
\]
\(\text { [Hint: See Example }(\mathrm{c})]\).
Using Corollary 2 in §2, and limit properties of the exponential and log functions, prove the "shorthand" Theorems \(11-16\) of §4.
Find \(\lim _{x \rightarrow+\infty}\left(1+\frac{1}{x}\right)^{\sqrt{x}}\).
Similarly, find a new solution of Problem 27 in Chapter 3, §15, reducing it to Problem \(26 .\)
Show that if \(f : E^{1} \rightarrow E^{*}\) has the Darboux property on \(B(\mathrm{e} . g ., \text { if } B \text { is }\) \(\text { convex and } f \text { is relatively continuous on } B)\) and if \(f\) is one to one on \(B\), then \(f\) is necessarily strictly monotone on \(B\).
Prove that if two real functions \(f, g\) are relatively continuous on \([a, b]\) \((a<b)\) and
\[
f(x) g(x)>0 \text { for } x \in[a, b] ,
\]
then the equation
\[
(x-a) f(x)+(x-b) g(x)=0
\]
has a solution between \(a\) and \(b ;\) similarly for the equation
\[
\frac{f(x)}{x-a}+\frac{g(x)}{x-b}=0 \quad\left(a, b \in E^{1}\right) .
\]
Similarly, discuss the solutions of
\[
\frac{2}{x-4}+\frac{9}{x-1}+\frac{1}{x-2}=0 .
\]