\documentclass{article} \usepackage{amssymb} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Created=Monday, June 25, 2001 17:45:57} %TCIDATA{LastRevised=Saturday, June 01, 2002 19:52:52} %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{Aplik\'{a}cie deriv\'{a}ci\'{\i}} \subsection{D\^{o}kaz vety o spojitosti inverznej funkcie} \textbf{D\^{o}kaz: }Pod\v{l}a \hyperref{vety}{}{}{M65.tex#10} je $% f:I\longrightarrow B$ r\'{y}dzomonot\'{o}nna. Nech $y_{0}\in B$ a $% f^{-1}\left( y_{0}\right) =x_{0}\in I$. V d\^{o}sledku r\'{y}dzej monot\'{o}% nnosti funkcie $f:I\longrightarrow B$ je $x_{0}$ vn\'{u}torn\'{y}m bodom intervalu $I$ ak $y_{0}$ je vn\'{u}torn\'{y}m bodom $B$. Nech $y_{0}$ je vn% \'{u}torn\'{y}m bodom intervalu $B$. Nech $O_{\varepsilon }\left( x_{0}\right) \subset I$ je \v{l}ubovo\v{l}n\'{e}. Preto\v{z}e $x_{0}$ je vn% \'{u}torn\'{y}m bodom $I$ existuje $O_{\eta }\left( x_{0}\right) \subset I\cap O_{\varepsilon }\left( x_{0}\right) $. Potom $f\left( O_{\eta }\left( x_{0}\right) \right) $ je interval, ktor\'{y} obsahuje bod $f(x_{0})=y_{0}$ ako vn\'{u}torn\'{y} bod. Preto existuje $O_{\delta }\left( y_{0}\right) $, tak\'{e} \v{z}e $O_{\delta }\left( y_{0}\right) \subset f\left( O_{\eta }\left( x_{0}\right) \right) \subset f\left( O_{\varepsilon }\left( x_{0}\right) \right) $. Odtia\v{l} \[ f^{-1}\left( O_{\delta }\left( y_{0}\right) \right) \subset \left( f^{-1}\circ f\right) \left( O_{\eta }\left( x_{0}\right) \right) =O_{\eta }\left( x_{0}\right) \subset O_{\varepsilon }\left( x_{0}\right) =O_{\varepsilon }\left( f^{-1}\left( y_{0}\right) \right) . \]% Teda $\forall O_{\varepsilon }\left( x_{0}\right) =O_{\varepsilon }\left( f^{-1}\left( y_{0}\right) \right) \,\exists O_{\delta }\left( y_{0}\right) $ tak, \v{z}e $f^{-1}\left( O_{\delta }\left( y_{0}\right) \right) \subset O_{\varepsilon }\left( x_{0}\right) =O_{\varepsilon }\left( f^{-1}\left( y_{0}\right) \right) $ teda $f^{-1}:B\longrightarrow I$ je spojit\'{a} v bode $y_{0}$. Ak $y_{0}$ je koncov\'{y} bod intervalu $B$, d\^{o}kaz je podobn\'{y}, preto ho ponech\'{a}vame na \v{c}itate\v{l}a. $\blacksquare $ \begin{center} \begin{tabular}{|c|} \hline \hyperref{{\small Sp\"{a}\v{t}}}{}{}{M65.tex#3} \\ \hline \end{tabular} \end{center} \end{document}