4.3.E: Problems on Continuity of Vector-Valued Functions
( \newcommand{\kernel}{\mathrm{null}\,}\)
Give an "ε,δ " proof of Theorem 1 for f±g.
[Hint: Proveed as in Theorem 1 of Chapter 3, §15, replacing max 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].