DarbouxsTheorem 2002-06-06 03:39:55 2004-02-25 05:11:28 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 \newcommand{\Prob}[2]{\mathbb{P}_{#1}\left\{#2\right\}} \newcommand{\norm}[1]{\left\|#1\right\|} % Some sets \newcommand{\Nats}{\mathbb{N}} \newcommand{\Ints}{\mathbb{Z}} \newcommand{\Reals}{\mathbb{R}} \newcommand{\Complex}{\mathbb{C}} Let $f:[a,b]\to\Reals$ be a real-valued continuous function on $[a,b]$, which is differentiable on $(a,b)$, differentiable from the right at $a$, and differentiable from the left at $b$. Then $f'$ \PMlinkescapetext{satisfies} the intermediate value theorem: for every $t$ between $f'_{+}(a)$ and $f'_{-}(b)$, there is some $x\in [a,b]$ such that $f'(x)=t$. Note that when $f$ is continuously differentiable ($f\in C^1([a,b])$), this is trivially true \emph{by} the intermediate value theorem. But even when $f'$ is \emph{not} continuous, Darboux's theorem places a severe restriction on what it can be.