4.8.E: Problems on Uniform Continuity; Continuity on Compact Sets
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.]
Do Problem 4 in Chapter 3, §17, and thus complete the last details in the proof of Theorem 4.
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 \((S, \rho)\).]
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 \(E^{1}\).]
Complete the missing details in Examples \((1)\) and \((2)\) and \((\mathrm{c})-(\mathrm{h})\).
Show that every polynomial of degree one on \(E^{n} (\text {*or } C^{n})\) is uniformly continuous.
Show that the arcsine function is uniformly continuous on \([-1,1] .\)
\(\text { [Hint: Use Example (d) and Theorems } 3 \text { and } 4 .]\)
\(\Rightarrow 8 .\) 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).)
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 \(B\).
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.
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 .]
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 \((\mathrm{b})\).]
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).
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 \((b)\).
Prove that if \(f\) is uniformly continuous on \(B,\) it is so on each subset \(A \subseteq B\).
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.
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 \(f\) \(\left.\text { on } F \text { follows from relative continuity on } F_{1} .\right]\)
\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 \(\text { points } 1,2, \ldots, m-1, \text { consider left and right limits.) Then use Theorems } 1-4 .]\)
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).