Saltar al contenido principal

# 4.3.E: Problemas sobre la continuidad de las funciones valoradas por vectores

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

## Ejercicio$$\PageIndex{1}$$

Dar una "$$\varepsilon, \delta$$" prueba de Teorema 1 para$$f \pm g$$.
[Pista: Se prueba como en el Teorema 1 del Capítulo 3, §15, sustituyendo$$\max \left(k^{\prime}, k^{\prime \prime}\right)$$ por$$\delta= \min \left(\delta^{\prime}, \delta^{\prime \prime}\right)$$. Así fijar$$\varepsilon>0$$ y$$p \in S .$$ Si$$f(x) \rightarrow q$$ y$$g(x) \rightarrow r$$ como$$x \rightarrow p$$ terminado$$B$$, entonces$$\left(\exists \delta^{\prime}, \delta^{\prime \prime}>0\right)$$ tal que
\ [
\ izquierda (\ forall x\ in B\ cap G_ {\ neg p}\ left (\ delta^ {\ prime}\ right)\ right)\ quad|f (x) -q|<\ frac {\ varepsilon} {2}\ text {y}\ left (\ forall x\ in B\ cap G_ {\ neg p}\ left (\ delta^ {\ prime\ prime}\ derecha)\ derecha)\ quad|g (x) -r|<\ frac {\ varepsilon} {2}.
\]
Poner$$\delta=\min \left(\delta^{\prime}, \delta^{\prime \prime}\right),$$ etc.$$]$$

En Problemas$$2,3,$$ y$$4, E=E^{n}(\text { * or another normed space }), F$$ es su campo escalar,$$B \subseteq A \subseteq(S, \rho),$$ y$$x \rightarrow p$$ más$$B .$$

## Ejercicio$$\PageIndex{2}$$

Para una función$$f : A \rightarrow E$$ probar que
\ [
f (x)\ rightarrow q\ LongLeftRightArrow|F (x) -q|\ rightarrow 0,
\]
\ [
\ begin {array} {l} {\ text {equivalentemente, iff} f (x) -q\ rightarrow\ overline {0}.}\\ {\ text {[Pista: Proceder como en el Capítulo} 3, 14 ,\ text {Corolario} 2.]}\ end {array}
\]

## Ejercicio$$\PageIndex{3}$$

Dado$$f : A \rightarrow\left(T, \rho^{\prime}\right),$$ con$$f(x) \rightarrow q$$ como$$x \rightarrow p$$ sobre$$B .$$ Mostrar que para algunos$$\delta>0, f$$ está acotado en$$B \cap G_{\neg p}(\delta),$$ i.e.,
\ [
f\ left [B\ cap G_ {\ neg p} (\ delta)\ right]\ text {es un conjunto acotado en}\ left (T,\ rho^ {\ prime}\ right).
\]
Así si$$T=E,$$ hay$$K \in E^{1}$$ tal que
\ [
\ left (\ forall x\ in B\ cap G_ {\ neg p} (\ delta)\ right)\ quad|f (x) |<K
\]
($$3, §13,$$Teorema del Capítulo 2$$)$$.

## Ejercicio$$\PageIndex{4}$$

Dado$$f, h : A \rightarrow E^{1}(C)$$ (o$$f : A \rightarrow E, h : A \rightarrow F ),$$ probar que si uno de$$f$$ y$$h$$ tiene límite 0 (respectivamente,$$\overline{0} ),$$ mientras que el otro está acotado en$$B \cap G_{\neg p}(\delta),$$ entonces$$h(x) f(x) \rightarrow 0(\overline{0})$$.

## Ejercicio$$\PageIndex{5}$$

Dado$$h : A \rightarrow E^{1}(C),$$ con$$h(x) \rightarrow a$$ como$$x \rightarrow p$$ terminado$$B,$$ y$$a \neq 0$$.
Demostrar que
\ [
(\ existe\ varepsilon,\ delta>0)\ left (\ forall x\ in B\ cap G_ {\ neg p} (\ delta)\ right)\ quad|h (x) |\ geq\ varepsilon,
\] es
decir,$$h(x)$$ está delimitado lejos de 0 on$$B \cap G_{\neg p}(\delta) .$$ De ahí mostrar que 1$$/ h$$ está acotado en $$B \cap G_{\neg p}(\delta) .$$
$$[\text { Hint: Proceed as in the proof of Corollary } 1 \text { in } §1, \text { with } q=a \text { and } r=0 . \text { Then use }$$
\ [
\ izquierda (\ forall x\ in B\ cap G_ {\ neg p} (\ delta)\ derecha)\ quad\ izquierda|\ frac {1} {h (x)}\ derecha|\ leq\ frac {1} {\ varepsilon}.]
\]

## Ejercicio$$\PageIndex{6}$$

Usando Problemas 1 a 5, dar una prueba independiente del Teorema 1.
[Pista: Proceder como en los Problemas 2 y 4 del Capítulo 3, §15 para obtener el Teorema 1 (ii). Luego usa Corolario 2 de$$\ 1 . ]$$

## Ejercicio$$\PageIndex{7}$$

Deducir los teoremas 1 y 2 del Capítulo$$3,$$ §15 a partir de los de la presente sección, ambientación$$A=B=N, S=E^{*},$$ y$$p=+\infty$$.
[Pista: Ver §1, Nota 5.]

## Ejercicio$$\PageIndex{8}$$

Rehacer el Problema 8 de §1 de dos maneras:
(i) Usar el Teorema 1 solamente.
(ii) Uso Teorema 3.
$$\left[\text { Example for }(\mathrm{i}) : \text { Find } \lim _{x \rightarrow 1}(x^{2}+1\right)$$.
Aquí$$f(x)=x^{2}+1,$$ o$$f=g g+h,$$ donde$$h(x)=1$$ (constante) y$$g(x)=x$$ (mapa de identidad). Como$$h$$ y$$g$$ son continuos$$(§1, \text { Examples }(\text { a }) \text { and }(\mathrm{b})),$$ así es$$f$$ por Teorema$$1 .$$ Así$$\lim _{x \rightarrow 1} f(x)=f(1)=1^{2}+1=2$$.
O, usando el Teorema 1$$(\text { ii) }, \lim _{x \rightarrow 1}\left(x^{2}+1\right)=\lim _{x \rightarrow 1} x^{2}+\lim _{x \rightarrow 1} 1, \text { etc. }]$$

## Ejercicio$$\PageIndex{9}$$

Definir$$f : E^{2} \rightarrow E^{1}$$ por
\ [
f (x, y) =\ frac {x^ {2} y} {\ left (x^ {4} +y^ {2}\ right)},\ text {con} f (0,0) =0.
\]
Mostrar que$$f(x, y) \rightarrow 0$$ como$$(x, y) \rightarrow(0,0)$$ a lo largo de cualquier línea recta a través$$\overline{0},$$ pero no por encima de la parábola$$y=x^{2}$$ (entonces el límite es$$\frac{1}{2} ) .$$ Deducir que$$f$$ es continuo$$\overline{0}=(0,0)$$ en adentro$$x$$ y$$y$$ por separado, pero no conjuntamente.

## Ejercicio$$\PageIndex{10}$$

Hacer problema$$9,$$ configurando
\ [
f (x, y) =0\ text {if} x=0,\ text {y} f (x, y) =\ frac {|y|} {x^ {2}}\ cdot 2^ {-|y|/x^ {2}}\ text {if} x\ neq 0.
\]

## Ejercicio$$\PageIndex{11}$$

Discutir la continuidad de$$f : E^{2} \rightarrow E^{1}$$ in$$x$$ y$$y$$ conjuntamente y por separado,$$\overline{0},$$ cuando
(
a)$$f(x, y)=\frac{x^{2} y^{2}}{x^{2}+y^{2}}, f(0,0)=0$$;
(b) parte$$f(x, y)=$$ integral de$$x+y$$;
(c)$$f(x, y)=x+\frac{x y}{|x|}$$ si; (d) si$$x \neq 0, f(0, y)=0$$;
(d)$$f(x, y)=\frac{x y}{|x|}+x \sin \frac{1}{y}$$ si $$x y \neq 0,$$y de$$f(x, y)=0$$ otra manera; e
)$$f(x, y)=\frac{1}{x} \sin \left(x^{2}+|x y|\right)$$ si$$x \neq 0,$$ y$$f(0, y)=0$$.
[Consejos: En$$(\mathrm{c}) \text { and }(\mathrm{d}),|f(x, y)| \leq|x|+|y| ; \text { in }(\mathrm{e}), \text { use }|\sin \alpha| \leq|\alpha| \cdot]$$.

4.3.E: Problemas sobre la continuidad de las funciones valoradas por vectores is shared under a CC BY 1.0 license and was authored, remixed, and/or curated by LibreTexts.