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Mathematics LibreTexts

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

  • Page ID
    23725
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    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).