%% This document created by Scientific Notebook (R) Version 3.5 %% Starting shell: article \documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=5.00.0.2570} %TCIDATA{} %TCIDATA{Created=Wednesday, February 10, 1999 13:29:48} %TCIDATA{LastRevised=Sunday, February 13, 2005 16:51:03} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=On line bluem.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{Counters=arabic,1} %TCIDATA{AllPages= %H=36 %F=36,\PARA{038
\QTR{small}{Matematick\U{e1} anal\U{fd}za II online - Funkcie, limita funkcie, spojit\U{e9} funkcie - Spojit\U{e9} funkcie\dotfill \thepage }} %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \begin{document} \author{A. U. Thor} \title{Lab Report} \date{The Date } \maketitle \begin{abstract} A Laboratory report created with Scientific Notebook \end{abstract} \section{Funkcie, limita funkcie, spojit\'{e} funkcie} \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maiindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maiindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{Ma2.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{Ma2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{Ma22.tex}}% %BeginExpansion \msihyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{Ma22.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O2.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{C2.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{C2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{Glos.tex}}% %BeginExpansion \msihyperref{Index}{}{}{Glos.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \subsection{Spojit\'{e} funkcie.} \begin{definition} Hovor\'{\i}me, \v{z}e funkcia $\mathbf{f}:A\left( \subset \mathbf{R}% ^{n}\right) \longrightarrow \mathbf{R}^{m}$ je \label{4}\emph{spojit\'{a} v bode} $\mathbf{a}\in A,$ ak pre \v{l}ubovo\v{l}n\'{e} okolie $O_{\varepsilon }\left( \mathbf{f}\left( \mathbf{a}\right) \right) $ bodu $\mathbf{f}\left( \mathbf{a}\right) $ existuje tak\'{e} $O_{\delta }\left( \mathbf{a}\right) ,$ \v{z}e $f\left( O_{\delta }\left( \mathbf{a}\right) \cap A\right) \subset O_{\varepsilon }\left( \mathbf{f}\left( \mathbf{a}\right) \right) .$ Teda : \[ \forall O_{\varepsilon }\left( \mathbf{f}\left( \mathbf{a}\right) \right) \,\exists O_{\delta }\left( \mathbf{a}\right) \,;\,f\left( O_{\delta }\left( \mathbf{a}\right) \cap A\right) \subset O_{\varepsilon }\left( \mathbf{f}% \left( \mathbf{a}\right) \right) . \]% Hovor\'{\i}me, \v{z}e $\mathbf{f}$ je \label{5}\emph{spojit\'{a} na} $% S\subset A,$ ak je spojit\'{a} v ka\v{z}dom bode $\mathbf{a}\in S.$ \end{definition} T\'{u}to defin\'{\i}ciu m\^{o}\v{z}eme prep\'{\i}sa\v{t} pomocou nerovnost% \'{\i} do nasleduj\'{u}ceho tvaru \[ \forall \varepsilon >0\,\exists \delta >0;\forall \mathbf{x}\in A;\,\,\left\| \mathbf{x-a}\right\| <\delta \Longrightarrow \left\| \mathbf{f}% \left( \mathbf{x}\right) -\mathbf{f}\left( \mathbf{a}\right) \right\| <\varepsilon . \] Uveden\'{a} defin\'{\i}cia sa podob\'{a} na defin\'{\i}ciu limity funkcie v bode. Rozdiel je v tom, \v{z}e v defin\'{\i}cii spojitosti funkcie v bode \emph{nepredpoklad\'{a}me}, \v{z}e bod $\mathbf{a}\in A$ je \emph{hromadn% \'{y} bod mno\v{z}iny} $A.$\ Ak bod $\mathbf{a}\in A$ je aj hromadn\'{y} bod mno\v{z}iny $A,$ potom z defin\'{\i}cie spojitosti funkcie v bode ihne\v{d} dost\'{a}vame tvrdenie: \begin{theorem} \label{6}(Veta o spojitosti funkcie v bode) Dan\'{a} je funkcia $\mathbf{f}% :A\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}^{m}\mathbf{% .}$ Nech$\mathbf{\,\ a}\in A$\ je hromadn\'{y} bod mno\v{z}iny $A.$ Potom $% \mathbf{f}$\ \ je spojit\'{a} v bode $\mathbf{a}$ vtedy a len vtedy\ ak $% \lim_{\mathbf{x}\longrightarrow \mathbf{a}}\mathbf{f}\left( \mathbf{x}% \right) $\ existuje a plat\'{\i} $\lim_{\mathbf{x}\longrightarrow \mathbf{a}}% \mathbf{f}\left( \mathbf{x}\right) =\mathbf{f}\left( \mathbf{a}\right) .$ \end{theorem} \begin{description} \item[Pozn\'{a}mka] Ak je funkcia $\mathbf{f}$\ definovan\'{a} v bode $% \mathbf{a}\in A,$ ktor\'{y} \textsl{nie je hromadn\'{y}m bodom }(tak\'{y} bod naz\'{y}vame izolovan\'{y}m bodom) jej defini\v{c}n\'{e}ho oboru (mno% \v{z}iny $A$\ ), vtedy m\^{o}\v{z}eme zvoli\v{t} $O_{\delta }\left( \mathbf{a% }\right) $ tak, \v{z}e $O_{\delta }\left( \mathbf{a}\right) \cap A=\left\{ \mathbf{a}\right\} .$ Teda pre ka\v{z}d\'{e} $x\in O_{\delta }\left( \mathbf{% a}\right) \cap A$ (je to len jedin\'{e} $\mathbf{x}$ a to $\mathbf{x}=% \mathbf{a}$) plat\'{\i} \[ \mathbf{f}\left( O_{\delta }\left( \mathbf{a}\right) \cap A\right) =\mathbf{f% }\left( \mathbf{a}\right) \in O_{\varepsilon }\left( \mathbf{f}\left( \mathbf{a}\right) \right) , \]% pre ka\v{z}d\'{e} $\varepsilon >0.$\ Teda $\mathbf{f}$ je v takom bode spojit% \'{a}. \item[Pozn\'{a}mka] Treba si uvedomi\v{t}, \v{z}e v $\varepsilon -\delta $ podmienke m\^{o}\v{z}e $\delta $ z\'{a}visie\v{t} od $\mathbf{a}$ aj od $% \varepsilon $, ako aj to, \v{z}e spojitos\v{t} je lok\'{a}lna vlastnos\v{t} z% \'{a}visiaca iba od chovania funkcie $\mathbf{f}$ v malom okol\'{\i} bodu $% \mathbf{a.}$ \end{description} \begin{example} Funkcie \[ f:\mathbf{R}^{n}\longrightarrow \mathbf{R},\,f\left( \mathbf{x}\right) =\left\| \mathbf{x}\right\| \]% a \[ \pi _{i}:\mathbf{R}^{n}\longrightarrow \mathbf{R},\,\pi _{i}\left( \mathbf{x}% \right) =x_{i},\,i=1,2,...,n \]% z pr\'{\i}kladov v predch\'{a}dzaj\'{u}cej \v{c}asti s\'{u} spojit\'{e} funkcie. \end{example} \begin{solution} Z %TCIMACRO{\hyperref{pr\'{\i}kladu}{}{}{Ma22.tex#8} }% %BeginExpansion \msihyperref{pr\'{\i}kladu}{}{}{Ma22.tex#8} %EndExpansion vieme, \v{z}e $\lim_{\mathbf{x}\longrightarrow \mathbf{a}}f\left( \mathbf{x}% \right) =\left\| \mathbf{a}\right\| =f\left( \mathbf{a}\right) ,\,\forall \mathbf{a}\in \mathbf{R}^{n}$ teda funkcia $f$ je spojit\'{a} v ka\v{z}dom bode $\mathbf{R}^{n}.$ Z %TCIMACRO{\hyperref{pr\'{\i}kladu}{}{}{Ma22.tex#9}}% %BeginExpansion \msihyperref{pr\'{\i}kladu}{}{}{Ma22.tex#9}% %EndExpansion , podobn\'{y}m sp\^{o}sobom dost\'{a}vame \[ \lim_{\mathbf{x}\longrightarrow \mathbf{a}}\pi _{i}\left( \mathbf{x}\right) =a_{i}=\pi _{i}\left( \mathbf{a}\right) ,\forall \mathbf{a}\in \mathbf{R}% ^{n},\,\forall \,i=1,2,...,n \]% t.j. ka\v{z}d\'{a} projekcia je spojit\'{a} v ka\v{z}dom bode z priestoru $% \mathbf{R}^{n}.$ \end{solution} Pre spojitos\v{t} m\^{o}\v{z}eme vyu\v{z}i\v{t} v\v{s}etky vety o limit\'{a}% ch, ktor\'{e} u\v{z} nebudeme dokazova\v{t}. \begin{theorem} Nech $\mathbf{f}:A\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}^{m}.$Potom s\'{u} nasleduj\'{u}ce tvrdenia ekvivalentn\'{e}: (i) $\mathbf{f}$ je spojit\'{a} v bode $\mathbf{a}\in A,$ (ii) pre ka\v{z}d\'{u} postupnos\v{t} $\left\{ \mathbf{x}^{\left( k\right) }\right\} _{k=1}^{\infty }\subset A,$ tak\'{u} \v{z}e $\lim_{k% \longrightarrow \infty }\mathbf{x}^{\left( k\right) }=\mathbf{% a\Longrightarrow }\lim_{k\longrightarrow \infty }\mathbf{f}\left( \mathbf{x}% ^{\left( k\right) }\right) =\mathbf{f}\left( \mathbf{a}\right) .$ \end{theorem} \begin{theorem} Nech $\mathbf{f}:A\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}^{m}$ m\'{a} komponenty $\mathbf{f}=(f_{1},f_{2},...,f_{m}).$ Potom $\mathbf{f}$ je spojit\'{a} v bode $\mathbf{a}\in A\Longleftrightarrow $ ak je ka\v{z}d\'{a} funkcia $f_{i},\,i=1,2,...,m$ spojit\'{a} v bode $% \mathbf{a.}$ \end{theorem} \begin{theorem} Nech $\mathbf{f,g}:A\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}^{m},$ s\'{u} spojit\'{e} v bode $\mathbf{a}\in A.$ Potom $\mathbf{% f+g},\,\alpha \mathbf{f}$ a pre $m=1$ aj $fg$ a ak $g\left( \mathbf{a}% \right) \neq 0,$ tak aj $\frac{f}{g}$ s\'{u} funkcie spojit\'{e} v bode $% \mathbf{a.}$ \end{theorem} \begin{theorem} Nech $\mathbf{f}:A\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}^{p}$ a $\mathbf{g}:B\left( \subset \mathbf{R}^{p}\right) \longrightarrow \mathbf{R}^{m}$ pri\v{c}om $\mathbf{f}(A)\subset B.$ Nech $% \mathbf{f}$ je spojit\'{a} v bode $\mathbf{a}\in A$ a $\mathbf{g}$ je spojit% \'{a} v bode $\mathbf{b=f}\left( \mathbf{a}\right) .$ Potom $\mathbf{g\circ f% }$ je spojit\'{a} v bode $\mathbf{a.}$ \end{theorem} \subsubsection{Spojitos\v{t} a kompaktnos\v{t}.} \begin{theorem} Nech $K\subset \mathbf{R}^{n}$ je kompaktn\'{a} mno\v{z}ina a nech $\mathbf{f% }:K\longrightarrow \mathbf{R}^{m}$ je funkcia spojit\'{a} na $K.$ Potom $% \mathbf{f}(K)$ je kompaktn\'{a} podmo\v{z}ina priestoru $\mathbf{R}^{m}.$ \end{theorem} \begin{tabular}{|c|} \hline {\small %TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO231.tex}}% %BeginExpansion \msihyperref{D\^{o}kaz}{}{}{DO231.tex}% %EndExpansion } \\ \hline \end{tabular}% \label{1} \begin{description} \item[Pozn\'{a}mka] Spojit\'{y} obraz uzavretej mno\v{z}iny nemus\'{\i} by% \v{t} uzavret\'{a} mno\v{z}ina a ani spojit\'{y} obraz ohrani\v{c}enej mno% \v{z}iny nemus\'{\i} by\v{t} ohrani\v{c}en\'{a} mno\v{z}ina. \end{description} \begin{example} a) Nech $f:\mathbf{R}\longrightarrow \mathbf{R},\,f\left( x\right) =\frac{% x^{2}}{x^{2}+1}.$ Potom $f(\mathbf{R})=\left\langle 0,1\right) $ nie je ani uzavret\'{a} ani otvoren\'{a}, b) Ak $f:(0,1)\longrightarrow \mathbf{R},f(x)=\frac{1}{x}.$ Potom $% f((0,1))=(1,\infty )$ nie je ohrani\v{c}en\'{a}. \end{example} \begin{definition} Funkcia $f:A\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}$ m\'{a} na mno\v{z}ine $A$ \label{7}\emph{maximum (minimum)} v bode $\mathbf{a% }\in A,$ak \[ f(\mathbf{x})\leq f(\mathbf{a}),\,\left( f(\mathbf{x})\geq f(\mathbf{a}% )\right) ,\,\forall \mathbf{x}\in A, \]% \v{c}o zapisujeme \[ \max_{\mathbf{x}\in A}f(\mathbf{x})=f(\mathbf{a}),\,(\min_{\mathbf{x}\in A}f(% \mathbf{x})=f(\mathbf{a})). \]% Hovor\'{\i}me, \v{z}e $f$ m\'{a} \label{8}\emph{lok\'{a}lne maximum (minimum)% } v bode $f(\mathbf{a}),$ ak existuje $O_{\delta }\left( \mathbf{a}\right) $ tak\'{e}, \v{z}e $f(\mathbf{x})\leq f(\mathbf{a})\;\left( f(\mathbf{x})\geq f(\mathbf{a})\right) ,\,\forall \mathbf{x}\in O_{\delta }\left( \mathbf{a}% \right) ,$ \v{c}o zapisujeme $\max_{\mathbf{x}\in O_{\delta }\left( \mathbf{a% }\right) }f(\mathbf{x})=f(\mathbf{a})$ $(\min_{\mathbf{x}\in O_{\delta }\left( \mathbf{a}\right) }f(\mathbf{x})=f(\mathbf{a})).$ \end{definition} Na \v{s}t\'{u}dium min\'{\i}m a max\'{\i}m ako je n\'{a}m u\v{z} zn\'{a}me z diferenci\'{a}lneho po\v{c}tu funkcie re\'{a}lnej premennej pou\v{z}\'{\i}% vame vyspelej\v{s}ie techniky. Teraz sformulujeme jednu existen\v{c}n\'{u} vetu. \begin{theorem} \label{3}Nech $f:K\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}$ je spojit\'{a} funkcia na kompaktnej mno\v{z}ine $K,$ potom $f$ m% \'{a} na $K$ maximum aj minimum. \end{theorem} \begin{tabular}{|c|} \hline {\small %TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO232.tex}}% %BeginExpansion \msihyperref{D\^{o}kaz}{}{}{DO232.tex}% %EndExpansion } \\ \hline \end{tabular}% \label{2} \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maiindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maiindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{Ma2.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{Ma2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{Ma22.tex}}% %BeginExpansion \msihyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{Ma22.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O2.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{C2.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{C2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{Glos.tex}}% %BeginExpansion \msihyperref{Index}{}{}{Glos.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za II} \section{Funkcie, limita funkcie, spojit\'{e} funkcie} \end{document}