4.9.E: Problems on the Darboux Property and Related Topics

Exercise $\PageIndex{1}$

Prove Note 1.

Exercise $\PageIndex{1'}$

Prove Note 3.

Exercise $\PageIndex{1''}$

Prove continuity at 0 in Example $(\mathrm{c})$.

Exercise $\PageIndex{2}$

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 $\left.B \text { in the theorem by the convex segment } L\left[\overline{p}_{i}, \overline{p}_{i+1}\right] .\right]$

Exercise $\PageIndex{3}$

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.

Exercise $\PageIndex{4}$

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?)]

Exercise $\PageIndex{5}$

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 $f(\overline{r})=c$.

Exercise $\PageIndex{6}$

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 $\left.x \in E^{1} .\right]$

Exercise $\PageIndex{7}$

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
$$h(x)=f(x)^{g(x)} .$$
$\text { [Hint: See Example }(\mathrm{c})]$.

Exercise $\PageIndex{8}$

Using Corollary 2 in §2, and limit properties of the exponential and log functions, prove the "shorthand" Theorems $11-16$ of §4.

Exercise $\PageIndex{8'}$

Find $\lim _{x \rightarrow+\infty}\left(1+\frac{1}{x}\right)^{\sqrt{x}}$.

Exercise $\PageIndex{8''}$

Similarly, find a new solution of Problem 27 in Chapter 3, §15, reducing it to Problem $26 .$

Exercise $\PageIndex{9}$

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 $B$.

Exercise $\PageIndex{10}$

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
$$\frac{f(x)}{x-a}+\frac{g(x)}{x-b}=0 \quad\left(a, b \in E^{1}\right) .$$

Exercise $\PageIndex{10'}$

Similarly, discuss the solutions of
$$\frac{2}{x-4}+\frac{9}{x-1}+\frac{1}{x-2}=0 .$$