4.3.E: Problems on Continuity of Vector-Valued Functions

Exercise $\PageIndex{1}$

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 $B .$

Exercise $\PageIndex{2}$

For a function $f : A \rightarrow E$ prove that
$$f(x) \rightarrow q \Longleftrightarrow|f(x)-q| \rightarrow 0 ,$$
$$\begin{array}{l}{\text { equivalently, iff } f(x)-q \rightarrow \overline{0} .} \\ {\text { [Hint: Proceed as in Chapter } 3, §14, \text { Corollary } 2 . ]}\end{array}$$

Exercise $\PageIndex{3}$

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$)$.

Exercise $\PageIndex{4}$

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 $h(x) f(x) \rightarrow 0(\overline{0})$.

Exercise $\PageIndex{5}$

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 $B \cap G_{\neg p}(\delta) .$
$[\text { Hint: Proceed as in the proof of Corollary } 1 \text { in } §1, \text { with } q=a \text { and } r=0 . \text { Then use }$
$$\left(\forall x \in B \cap G_{\neg p}(\delta)\right) \quad\left|\frac{1}{h(x)}\right| \leq \frac{1}{\varepsilon} . ]$$

Exercise $\PageIndex{6}$

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 $\$ 1 . ]$

Exercise $\PageIndex{7}$

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.]

Exercise $\PageIndex{8}$

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$(\text { ii) }, \lim _{x \rightarrow 1}\left(x^{2}+1\right)=\lim _{x \rightarrow 1} x^{2}+\lim _{x \rightarrow 1} 1, \text { etc. }]$

Exercise $\PageIndex{9}$

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.

Exercise $\PageIndex{10}$

Do Problem $9,$ setting
$$f(x, y)=0 \text { if } x=0, \text { and } f(x, y)=\frac{|y|}{x^{2}} \cdot 2^{-|y| / x^{2}} \text { if } x \neq 0 .$$

Exercise $\PageIndex{11}$

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 $(\mathrm{c}) \text { and }(\mathrm{d}),|f(x, y)| \leq|x|+|y| ; \text { in }(\mathrm{e}), \text { use }|\sin \alpha| \leq|\alpha| \cdot]$.