total variation TotalVariation 2003-02-08 13:35:32 2008-03-27 12:26:58 Definition bounded variation rectifiable path % 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 Let $\gamma:[a,b]\rightarrow X$ be a function mapping an interval $[a,b]$ to a metric space $(X,d)$. We say that $\gamma$ is of \emph{bounded variation} if there is a constant $M$ such that, for each partition $P=\{a=t_0<t_1<\cdots < t_n=b\}$ of $[a,b]$, \[ v(\gamma, P)= \sum_{k=1}^n d(\gamma(t_k),\gamma(t_{k-1})) \leq M. \] The \emph{total variation} $V_\gamma$ of $\gamma$ is defined by \[V_\gamma = \sup\{v(\gamma,P):\textnormal{$P$ is a partition of $[a,b]$}\}.\] It can be shown that, if $X$ is either $\mathbb{R}$ or $\mathbb{C}$, every continuously differentiable (or piecewise continuously differentiable) function $\gamma:[a,b]\rightarrow X$ \PMlinkname{is of bounded variation}{ContinuousDerivativeImpliesBoundedVariation}, and \[V_\gamma = \int_a^b |\gamma'(t)|dt.\] Also, if $\gamma$ is of bounded variation and $f:[a,b]\rightarrow X$ is continuous, then the Riemann-Stieltjes integral $\int_a^b fd\gamma$ is finite. If $\gamma$ is also continuous, it is said to be a \emph{rectifiable path}, and $V(\gamma)$ is the length of its trace. If $X=\mathbb{R}$, it can be shown that $\gamma$ is of bounded variation if and only if it is the difference of two monotonic functions.