Bolzano's theorem BolzanosTheorem 2006-02-02 12:39:09 2006-02-02 13:13:16 Theorem intermediate value 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 \theoremstyle{definition} \newtheorem*{thmplain}{Theorem} {\em A continuous function can not change its \PMlinkname{sign}{SignumFunction} without going through the zero.} This contents of Bolzano's theorem may be formulated more precisely as the \begin{thmplain} If a real function $f$ is continuous on a closed interval $I$ and the values of $f$ in the end points of $I$ have \PMlinkname{opposite}{Positive} signs, then there exists a zero of this function inside the interval. \end{thmplain} The theorem is used when using the interval halving method for getting an approximate value of a root of an equation of the form\, $f(x) = 0$.