
# 4.8.E: Problems on Uniform Continuity; Continuity on Compact Sets


Exercise $$\PageIndex{1}$$

Prove that if $$f$$ is relatively continuous on each compact subset of $$D,$$ then it is relatively continuous on $$D .$$
[Hint: Use Theorem 1 of §2 and Problem 7 in §6.]

Exercise $$\PageIndex{2}$$

Do Problem 4 in Chapter 3, §17, and thus complete the last details in the proof of Theorem 4.

Exercise $$\PageIndex{3}$$

Give an example of a continuous one-to-one map $$f$$ such that $$f^{-1}$$ is not continuous.
[Hint: Show that any map is continuous on a discrete space $$(S, \rho)$$.]

Exercise $$\PageIndex{4}$$

Give an example of a continuous function $$f$$ and a compact set $$D \subseteq$$ $$\left(T, \rho^{\prime}\right)$$ such that $$f^{-1}[D]$$ is not compact.
[Hint: Let $$f$$ be constant on $$E^{1}$$.]

Exercise $$\PageIndex{5}$$

Complete the missing details in Examples $$(1)$$ and $$(2)$$ and $$(\mathrm{c})-(\mathrm{h})$$.

Exercise $$\PageIndex{6}$$

Show that every polynomial of degree one on $$E^{n} (\text {*or } C^{n})$$ is uniformly continuous.

Exercise $$\PageIndex{7}$$

Show that the arcsine function is uniformly continuous on $$[-1,1] .$$
$$\text { [Hint: Use Example (d) and Theorems } 3 \text { and } 4 .]$$

Exercise $$\PageIndex{8}$$

$$\Rightarrow 8 .$$ Prove that if $$f$$ is uniformly continuous on $$B,$$ and if $$\left\{x_{m}\right\} \subseteq B$$ is a Cauchy sequence, so is $$\left\{f\left(x_{m}\right)\right\} .$$ (Briefly, $$f$$ preserves Cauchy sequences.) Show that this may fail if $$f$$ is only continuous in the ordinary sense. (See Example (h).)

Exercise $$\PageIndex{9}$$

Prove that if $$f : S \rightarrow T$$ is uniformly continuous on $$B \subseteq S,$$ and $$g : T \rightarrow U$$ is uniformly continuous on $$f[B],$$ then the composite function $$g \circ f$$ is uniformly continuous on $$B$$.

Exercise $$\PageIndex{10}$$

Show that the functions $$f$$ and $$f^{-1}$$ in Problem 5 of Chapter 3, §11 are contraction maps, 5 hence uniformly continuous. By Theorem 1, find again that $$\left(E^{*}, \rho^{\prime}\right)$$ is compact.

Exercise $$\PageIndex{11}$$

Let $$A^{\prime}$$ be the set of all cluster points of $$A \subseteq(S, \rho) .$$ Let $$f : A \rightarrow\left(T, \rho^{\prime}\right)$$ be uniformly continuous on $$A,$$ and let $$\left(T, \rho^{\prime}\right)$$ be complete.
(i) Prove that $$\lim _{x \rightarrow p} f(x)$$ exists at each $$p \in A^{\prime}$$ .
(ii) Thus define $$f(p)=\lim _{x \rightarrow p} f(x)$$ for each $$p \in A^{\prime}-A,$$ and show
that $$f$$ so extended is uniformly continuous on the set $$\overline{A}=A \cup A^{\prime} .$$
(iii) Consider, in particular, the case $$A=(a, b) \subseteq E^{1},$$ so that
$\overline{A}=A^{\prime}=[a, b] .$
[Hint: Take any sequence $$\left\{x_{m}\right\} \subseteq A, x_{m} \rightarrow p \in A^{\prime} .$$ As it is Cauchy (why?), so is $$\left\{f\left(x_{m}\right)\right\}$$ by Problem $$8 .$$ Use Corollary 1 in §2 to prove existence of $$\lim _{x \rightarrow p} f(x)$$. For uniform continuity, use definitions; in case (iii), use Theorem 4 .]

Exercise $$\PageIndex{12}$$

Prove that if two functions $$f, g$$ with values in a normed vector space are uniformly continuous on a set $$B,$$ so also are $$f \pm g$$ and $$a f$$ for a fixed scalar $$a .$$
For real functions, prove this also for $$f \vee g$$ and $$f \wedge g$$ defined by
$(f \vee g)(x)=\max (f(x), g(x))$
and
$(f \wedge g)(x)=\min (f(x), g(x)) .$
[Hint: After proving the first statements, verify that
$\max (a, b)=\frac{1}{2}(a+b+|b-a|) \text { and } \min (a, b)=\frac{1}{2}(a+b-|b-a|)$
and use Problem 9 and Example $$(\mathrm{b})$$.]

Exercise $$\PageIndex{13}$$

Let $$f$$ be vector valued and $$h$$ scalar valued, with both uniformly continuous on $$B \subseteq(S, \rho) .$$
Prove that
(i) if $$f$$ and $$h$$ are bounded on $$B$$ , then $$h f$$ is uniformly continuous on $$B$$;
(ii) the function $$f / h$$ is uniformly continuous on $$B$$ if $$f$$ is bounded on $$B$$ and $$h$$ is "bounded away" from 0 on $$B$$ , i.e.,
$(\exists \delta>0)(\forall x \in B) \quad|h(x)| \geq \delta .$
Give examples to show that without these additional conditions, $$h f$$ and $$f / h$$ may not be uniformly continuous (see Problem 14 below).

Exercise $$\PageIndex{14}$$

In the following cases, show that $$f$$ is uniformly continuous on $$B \subseteq E^{1}$$, but only continuous (in the ordinary sense) on $$D,$$ as indicated, with $$0<a<b<+\infty$$.
(a) $$f(x)=\frac{1}{x^{2}} ; B=[a,+\infty) ; D=(0,1)$$.
(b) $$f(x)=x^{2} ; B=[a, b] ; D=[a,+\infty)$$.
(c) $$f(x)=\sin \frac{1}{x} ; B$$ and $$D$$ as in $$(a)$$.
(d) $$f(x)=x \cos x ; B$$ and $$D$$ as in $$(b)$$.

Exercise $$\PageIndex{15}$$

Prove that if $$f$$ is uniformly continuous on $$B,$$ it is so on each subset $$A \subseteq B$$.

Exercise $$\PageIndex{16}$$

For nonvoid sets $$A, B \subseteq(S, \rho),$$ define
$\rho(A, B)=\inf \{\rho(x, y) | x \in A, y \in B\} .$
Prove that if $$\rho(A, B)>0$$ and if $$f$$ is uniformly continuous on each of $$A$$ and $$B,$$ it is so on $$A \cup B$$.
Show by an example that this fails if $$\rho(A, B)=0,$$ even if $$A \cap B=\emptyset$$ $$(\mathrm{e} . g ., \text { take } A=[0,1], B=(1,2] \text { in } E^{1}, \text { making } f \text { constant on each of } A$$ $$\text { and } B)$$.
Note, however, that if $$A$$ and $$B$$ are compact, $$A \cap B=\emptyset$$ implies $$\rho(A, B)>0 . \text { (Prove it using Problem } 13 \text { in } §6 .)$$ Thus $$A \cap B=\emptyset$$ suffices in this case.

Exercise $$\PageIndex{17}$$

Prove that if $$f$$ is relatively continuous on each of the disjoint closed sets
$F_{1}, F_{2}, \ldots, F_{n} ,$
it is relatively continuous on their union
$F=\bigcup_{k=1}^{n} F_{k} ;$
hence (see Problem 6 of §6) it is uniformly continuous on $$F$$ if the $$F_{k}$$ are compact.
[Hint: Fix any $$p \in F .$$ Then $$p$$ is in some $$F_{k},$$ say, $$p \in F_{1} .$$ As the $$F_{k}$$ are disjoint, $$p \notin F_{2}, \ldots, F_{p} ;$$ hence $$p$$ also is no cluster point of any of $$F_{2}, \ldots, F_{n}$$ (for they are closed).
Deduce that there is a globe $$G_{p}(\delta)$$ disjoint from each of $$F_{2}, \ldots, F_{n},$$ so that $$F \cap G_{p}(\delta)=F_{1} \cap G_{p}(\delta) .$$ From this it is easy to show that relative continuity of $$f$$ $$\left.\text { on } F \text { follows from relative continuity on } F_{1} .\right]$$

Exercise $$\PageIndex{18}$$

\Rightarrow 18 .\) Let $$\overline{p}_{0}, \overline{p}_{1}, \ldots, \overline{p}_{m}$$ be fixed points in $$E^{n} (^{*}$$ or in another normed space).
Let
$f(t)=\overline{p}_{k}+(t-k)\left(\overline{p}_{k+1}-\overline{p}_{k}\right)$
whenever $$k \leq t \leq k+1, t \in E^{1}, k=0,1, \ldots, m-1$$.
Show that this defines a uniformly continuous mapping $$f$$ of the interval $$[0, m] \subseteq E^{1}$$ onto the "polygon"
$\bigcup_{k=0}^{m-1} L [ p_{k}, p_{k+1} ] .$
In what case is $$f$$ one to one? Is $$f^{-1}$$ uniformly continuous on each $$L\left[p_{k}, p_{k+1}\right] ?$$ On the entire polygon?
[Hint: First prove ordinary continuity on $$[0, m]$$ using Theorem 1 of §3. (For the $$\text { points } 1,2, \ldots, m-1, \text { consider left and right limits.) Then use Theorems } 1-4 .]$$

Exercise $$\PageIndex{19}$$

Prove the sequential criterion for uniform continuity: A function $$f : A \rightarrow T$$ is uniformly continuous on a set $$B \subseteq A$$ iff for any two (not necessarily convergent) sequences $$\left\{x_{m}\right\}$$ and $$\left\{y_{m}\right\}$$ in $$B,$$ with $$\rho\left(x_{m}, y_{m}\right) \rightarrow 0,$$ we have $$\rho^{\prime}\left(f\left(x_{m}\right), f\left(y_{m}\right)\right) \rightarrow 0$$ (i.e., $$f$$ preserves con-current pairs of sequences; see Problem 4 in Chapter 3, §17).