SardsTheorem2 2006-08-31 20:44:18 2006-10-06 14:25:25 Theorem % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newtheorem*{thm}{Theorem} \begin{thm} Let $U$ be an open set in $R^m$, $V$ be an open set in $R^n$ and $f: U \to V$ be a $C^k$ function. If $k \geq m/n$, then the set of critical values of $f$ has $n$-dimensional measure zero. \end{thm} \begin{thebibliography}{99} \bibitem{kr} S.G. Krantz, H. R. Parks, \emph{The Implicit Function Theorem: History, Theory, and Applications}, Birkh\"auser, Boston, c. 2002 \bibitem{fe} H. Federer, \emph{Geometric Measure Theory}, Springer-Verlag, New York, 1969 \end{thebibliography}