\documentclass{article} \usepackage{amssymb} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Created=Monday, June 25, 2001 17:45:57} %TCIDATA{LastRevised=Saturday, June 01, 2002 19:30:37} %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 I 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{Spojitos\v{t} funkcie v bode} \subsection{D\^{o}kazy viet o vlastnostiach spojit\'{y}ch funkci\'{\i}} \textbf{D\^{o}kaz: }Nech $\varepsilon >0$. Preto\v{z}e $f$ je spojit\'{a} na $\left\langle a,b\right\rangle $, $\forall t\in \left\langle a,b\right\rangle \;\exists \delta \left( t\right) >0$ tak\'{e}, \v{z}e ak $% x\in \left\langle a,b\right\rangle :\left| t-x\right| <\delta \left( t\right) \Longrightarrow |f(x)-f(t)|<\frac{\varepsilon }{2}$. Potom syst\'{e}% m otvoren\'{y}ch intervalov \[ C=\left\{ \left( t-\frac{\delta \left( t\right) }{2},t+\frac{\delta \left( t\right) }{2}\right) :t\in \left\langle a,b\right\rangle \right\} \] pokr\'{y}va $\left\langle a,b\right\rangle $. Existuje teda kone\v{c}n\'{e} podpokrytie \[ C_{0}=\left\{ \left( t_{1}-\frac{\delta \left( t_{1}\right) }{2},t_{1}+\frac{% \delta \left( t_{1}\right) }{2}\right) ,...,\left( t_{n}-\frac{\delta \left( t_{n}\right) }{2},t_{n}+\frac{\delta \left( t_{n}\right) }{2}\right) \right\} \] a nech $\delta =\min \left\{ \frac{\delta \left( t_{1}\right) }{2},...\frac{% \delta \left( t_{n}\right) }{2}\right\} $. Potom ak $x\in \left\langle a,b\right\rangle $ mus\'{\i} $x$ le\v{z}a\v{t} v jednom z intervalov $\left( t_{k}-\frac{\delta \left( t_{k}\right) }{2},t_{k}+\frac{\delta \left( t_{k}\right) }{2}\right) $, potom $|x-t_{k}|<\frac{\delta \left( t_{k}\right) }{2}$. Ak $y\in \left\langle a,b\right\rangle $ a $|y-x|<\delta $, tak $|y-t_{k}|=|y-x+x-t_{k}|\leq |y-x|+|x-t_{k}|<\delta +\frac{\delta \left( t_{k}\right) }{2}<\delta \left( t_{k}\right) $ odkia\v{l} \[ |f(x)-f(y)|=|f(x)-f(t_{k})+f(t_{k})-f(y)|\leq |f(x)-f(t_{k})|+|f(t_{k})-f(y)|<\frac{\varepsilon }{2}+\frac{\varepsilon }{2}% =\varepsilon , \] Teda $f$ je rovnomerne spojit\'{a} na $\left\langle a,b\right\rangle $. $% \blacksquare $ \begin{center} \begin{tabular}{|c|} \hline \hyperref{{\small Sp\"{a}\v{t}}}{}{}{M42.tex#7} \\ \hline \end{tabular} \end{center} \end{document}