4.3.E: Problems on Continuity of Vector-Valued Functions
Give an "\(\varepsilon, \delta\) " proof of Theorem 1 for \(f \pm g\).
[Hint: Proveed as in Theorem 1 of Chapter 3, §15, replacing \(\max \left(k^{\prime}, k^{\prime \prime}\right)\) by \(\delta= \min \left(\delta^{\prime}, \delta^{\prime \prime}\right)\). Thus fix \(\varepsilon>0\) and \(p \in S .\) If \(f(x) \rightarrow q\) and \(g(x) \rightarrow r\) as \(x \rightarrow p\) over \(B\), then \(\left(\exists \delta^{\prime}, \delta^{\prime \prime}>0\right)\) such that
\[
\left(\forall x \in B \cap G_{\neg p}\left(\delta^{\prime}\right)\right) \quad|f(x)-q|<\frac{\varepsilon}{2} \text { and }\left(\forall x \in B \cap G_{\neg p}\left(\delta^{\prime \prime}\right)\right) \quad|g(x)-r|<\frac{\varepsilon}{2} .
\]
Put \(\delta=\min \left(\delta^{\prime}, \delta^{\prime \prime}\right),\) etc. \(]\)
In Problems \(2,3,\) and \(4, E=E^{n}(\text { * or another normed space }), F\) is its scalar field, \(B \subseteq A \subseteq(S, \rho),\) and \(x \rightarrow p\) over \(B .\)
For a function \(f : A \rightarrow E\) prove that
\[
f(x) \rightarrow q \Longleftrightarrow|f(x)-q| \rightarrow 0 ,
\]
\[
\begin{array}{l}{\text { equivalently, iff } f(x)-q \rightarrow \overline{0} .} \\ {\text { [Hint: Proceed as in Chapter } 3, §14, \text { Corollary } 2 . ]}\end{array}
\]
Given \(f : A \rightarrow\left(T, \rho^{\prime}\right),\) with \(f(x) \rightarrow q\) as \(x \rightarrow p\) over \(B .\) Show that for some \(\delta>0, f\) is bounded on \(B \cap G_{\neg p}(\delta),\) i.e.,
\[
f\left[B \cap G_{\neg p}(\delta)\right] \text { is a bounded set in }\left(T, \rho^{\prime}\right) .
\]
Thus if \(T=E,\) there is \(K \in E^{1}\) such that
\[
\left(\forall x \in B \cap G_{\neg p}(\delta)\right) \quad|f(x)|<K
\]
(Chapter \(3, §13,\) Theorem 2\()\).
Given \(f, h : A \rightarrow E^{1}(C)\) (or \(f : A \rightarrow E, h : A \rightarrow F ),\) prove that if one of \(f\) and \(h\) has limit 0 (respectively, \(\overline{0} ),\) while the other is bounded on \(B \cap G_{\neg p}(\delta),\) then \(h(x) f(x) \rightarrow 0(\overline{0})\).
Given \(h : A \rightarrow E^{1}(C),\) with \(h(x) \rightarrow a\) as \(x \rightarrow p\) over \(B,\) and \(a \neq 0\).
Prove that
\[
(\exists \varepsilon, \delta>0)\left(\forall x \in B \cap G_{\neg p}(\delta)\right) \quad|h(x)| \geq \varepsilon ,
\]
i.e., \(h(x)\) is bounded away from 0 on \(B \cap G_{\neg p}(\delta) .\) Hence show that 1\(/ h\) is bounded on \(B \cap G_{\neg p}(\delta) .\)
\([\text { Hint: Proceed as in the proof of Corollary } 1 \text { in } §1, \text { with } q=a \text { and } r=0 . \text { Then use }\)
\[
\left(\forall x \in B \cap G_{\neg p}(\delta)\right) \quad\left|\frac{1}{h(x)}\right| \leq \frac{1}{\varepsilon} . ]
\]
Using Problems 1 to 5, give an independent proof of Theorem 1.
[Hint: Proceed as in Problems 2 and 4 of Chapter 3, §15 to obtain Theorem 1(ii). Then use Corollary 2 of \(\$ 1 . ]\)
Deduce Theorems 1 and 2 of Chapter \(3,\) §15 from those of the present section, setting \(A=B=N, S=E^{*},\) and \(p=+\infty\).
[Hint: See §1, Note 5.]
Redo Problem 8 of §1 in two ways:
(i) Use Theorem 1 only.
(ii) Use Theorem 3.
\(\left[\text { Example for }(\mathrm{i}) : \text { Find } \lim _{x \rightarrow 1}(x^{2}+1\right)\).
Here \(f(x)=x^{2}+1,\) or \(f=g g+h,\) where \(h(x)=1\) (constant) and \(g(x)=x\) (identity map). As \(h\) and \(g\) are continuous \((§1, \text { Examples }(\text { a }) \text { and }(\mathrm{b})),\) so is \(f\) by Theorem \(1 .\) Thus \(\lim _{x \rightarrow 1} f(x)=f(1)=1^{2}+1=2\).
Or, using Theorem 1\((\text { ii) }, \lim _{x \rightarrow 1}\left(x^{2}+1\right)=\lim _{x \rightarrow 1} x^{2}+\lim _{x \rightarrow 1} 1, \text { etc. }]\)
Define \(f : E^{2} \rightarrow E^{1}\) by
\[
f(x, y)=\frac{x^{2} y}{\left(x^{4}+y^{2}\right)}, \text { with } f(0,0)=0 .
\]
Show that \(f(x, y) \rightarrow 0\) as \((x, y) \rightarrow(0,0)\) along any straight line through \(\overline{0},\) but not over the parabola \(y=x^{2}\) (then the limit is \(\frac{1}{2} ) .\) Deduce that \(f\) is continuous at \(\overline{0}=(0,0)\) in \(x\) and \(y\) separately, but not jointly.
Do Problem \(9,\) setting
\[
f(x, y)=0 \text { if } x=0, \text { and } f(x, y)=\frac{|y|}{x^{2}} \cdot 2^{-|y| / x^{2}} \text { if } x \neq 0 .
\]
Discuss the continuity of \(f : E^{2} \rightarrow E^{1}\) in \(x\) and \(y\) jointly and separately,
at \(\overline{0},\) when
(a) \(f(x, y)=\frac{x^{2} y^{2}}{x^{2}+y^{2}}, f(0,0)=0\);
(b) \(f(x, y)=\) integral part of \(x+y\);
(c) \(f(x, y)=x+\frac{x y}{|x|}\) if \(x \neq 0, f(0, y)=0\);
(d) \(f(x, y)=\frac{x y}{|x|}+x \sin \frac{1}{y}\) if \(x y \neq 0,\) and \(f(x, y)=0\) otherwise;
(e) \(f(x, y)=\frac{1}{x} \sin \left(x^{2}+|x y|\right)\) if \(x \neq 0,\) and \(f(0, y)=0\).
[Hints: In \((\mathrm{c}) \text { and }(\mathrm{d}),|f(x, y)| \leq|x|+|y| ; \text { in }(\mathrm{e}), \text { use }|\sin \alpha| \leq|\alpha| \cdot]\).