\documentclass{article} \usepackage{amssymb} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Created=Monday, June 25, 2001 17:45:57} %TCIDATA{LastRevised=Monday, March 24, 2003 12:42:24} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=On line bluem.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{AllPages= %H=36 %F=36,\PARA{038
\QTR{small}{Matematick\U{e1} anal\U{fd}za II online - D\U{f4}kazy\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{Limita funkcie} \subsection{Veta o limite zlo\v{z}enej funkcie} \textbf{D\^{o}kaz: }M\'{a}me $\lim_{\mathbf{y}\longrightarrow \mathbf{c}}% \mathbf{g}\left( \mathbf{x}\right) =\mathbf{b,}$ t.j. $\forall \varepsilon >0\,\exists \eta >0;\,\mathbf{y}\in B;\,0<\left\| \mathbf{y}-\mathbf{c}% \right\| <\eta \Longrightarrow \left\| \mathbf{g}\left( \mathbf{y}\right) -% \mathbf{b}\right\| <\varepsilon $ a preto\v{z}e $\lim_{\mathbf{x}% \longrightarrow \mathbf{a}}\mathbf{f}\left( \mathbf{x}\right) =\mathbf{c}$ tak $\forall \chi >0\,\exists \delta >0;\,\mathbf{x}\in A;\,0<\left\| \mathbf{x}-\mathbf{a}\right\| <\delta \Longrightarrow \left\| \mathbf{f}% \left( \mathbf{x}\right) -\mathbf{c}\right\| <\chi .$ M\'{a}me $\mathbf{f}(% \mathbf{x})\neq \mathbf{c}.$ Potom ku $\eta =\chi $ m\^{o}\v{z}eme n\'{a}js% \v{t} $\delta >0$ tak\'{e}, \v{z}e ak $\mathbf{x}\in A;\,0<\left\| \mathbf{x}% -\mathbf{a}\right\| <\delta \Longrightarrow \left\| \mathbf{f}\left( \mathbf{% x}\right) -\mathbf{c}\right\| <\eta ,$ t.j. \[ \forall \varepsilon >0\,\exists \delta >0;\,\mathbf{x}\in A;\,0<\left\| \mathbf{x}-\mathbf{a}\right\| <\delta \Longrightarrow \left\| \left( \mathbf{% g\circ f}\right) \left( \mathbf{x}\right) -\mathbf{b}\right\| <\varepsilon .\blacksquare \] \begin{center} \begin{tabular}{|c|} \hline {\small \hyperref{Sp\"{a}\v{t}}{}{}{Ma22.tex#6}} \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za II} \end{document}