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26A06-GeneralizedIntermediateValueTheorem.tex
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26A06-GeneralizedIntermediateValueTheorem.tex
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\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{GeneralizedIntermediateValueTheorem}
\pmcreated{2013-03-22 17:17:44}
\pmmodified{2013-03-22 17:17:44}
\pmowner{azdbacks4234}{14155}
\pmmodifier{azdbacks4234}{14155}
\pmtitle{generalized intermediate value theorem}
\pmrecord{8}{39639}
\pmprivacy{1}
\pmauthor{azdbacks4234}{14155}
\pmtype{Theorem}
\pmcomment{trigger rebuild}
\pmclassification{msc}{26A06}
%\pmkeywords{continuous}
%\pmkeywords{connected}
%\pmkeywords{order}
%\pmkeywords{order topology}
\pmrelated{OrderTopology}
\pmrelated{TotalOrder}
\pmrelated{Continuous}
\pmrelated{ConnectedSpace}
\pmrelated{ConnectednessIsPreservedUnderAContinuousMap}
\endmetadata
%packages
\usepackage{amsmath,mathrsfs,amsfonts,amsthm}
%theorem environments
\theoremstyle{plain}
\newtheorem*{thm*}{Theorem}
\newtheorem*{lem*}{Lemma}
\newtheorem*{cor*}{Corollary}
\newtheorem*{prop*}{Proposition}
%delimiters
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\begin{document}
\begin{thm*}
Let $f:X\rightarrow Y$ be a continuous function with $X$ a connected space and $Y$ a totally ordered set in the order topology. If $x_1,x_2\in X$ and $y\in Y$ lies between $f(x_1)$ and $f(x_2)$, then there exists $x\in X$ such that $f(x)=y$.
\end{thm*}
\begin{proof}
The sets $U=f(X)\cap(-\infty,y)$ and $V=f(X)\cap(y,\infty)$ are disjoint open subsets of $f(X)$ in the subspace topology, and they are both non-empty, as $f(x_1)$ is contained in one and $f(x_2)$ is contained in the other. If $y\notin f(X)$, then $U\cup V$ constitutes a \PMlinkescapetext{separation} of the space $f(X)$, contradicting the hypothesis that $f(X)$ is the continuous image of the connected space $X$. Thus there must exist $x\in X$ such that $f(x)=y$.
\end{proof}
This version of the intermediate value theorem reduces to the familiar one of \PMlinkid{real analysis}{7599} when $X$ is taken to be a closed interval in $\mathbb{R}$ and $Y$ is taken to be $\mathbb{R}$.
\begin{thebibliography}{1}
\bibitem{munkres}
J. Munkres, \emph{Topology}, 2nd ed. Prentice Hall, 1975.
\end{thebibliography}
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\end{document}