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Hometotal variation

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# total variation

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 *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 *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 $\gamma:[a,b]\rightarrow X$ is of bounded variation, and

$V_{\gamma}=\int_{a}^{b}|\gamma^{{\prime}}(t)|dt.$ |

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 *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.

## Mathematics Subject Classification

26A45*no label found*26B30

*no label found*

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