# total variation

Let $\gamma :[a,b]\to X$ 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
$$ of $[a,b]$,

$$v(\gamma ,P)=\sum _{k=1}^{n}d(\gamma ({t}_{k}),\gamma ({t}_{k-1}))\le M.$$ |

The *total variation ^{}* ${V}_{\gamma}$ of $\gamma $ is defined by

$${V}_{\gamma}=sup\{v(\gamma ,P):P\text{is a partition of}[a,b]\}.$$ |

It can be shown that, if $X$ is either $\mathbb{R}$ or $\u2102$, every continuously differentiable (or piecewise continuously differentiable) function $\gamma :[a,b]\to X$ is of bounded variation (http://planetmath.org/ContinuousDerivativeImpliesBoundedVariation), and

$${V}_{\gamma}={\int}_{a}^{b}|{\gamma}^{\prime}(t)|\mathit{d}t.$$ |

Also, if $\gamma $ is of bounded variation and $f:[a,b]\to X$ is continuous^{}, then the Riemann-Stieltjes integral ${\int}_{a}^{b}f\mathit{d}\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.

Title | total variation |

Canonical name | TotalVariation |

Date of creation | 2013-03-22 13:26:09 |

Last modified on | 2013-03-22 13:26:09 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 8 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 26A45 |

Classification | msc 26B30 |

Related topic | BVFunction |

Related topic | IntegralRepresentationOfLengthOfSmoothCurve |

Related topic | OscillationOfAFunction |

Defines | bounded variation |

Defines | rectifiable path |