total variation

Let γ:[a,b]X be a function mapping an interval [a,b] to a metric space (X,d). We say that γ is of bounded variationMathworldPlanetmath if there is a constant M such that, for each partition P={a=t0<t1<<tn=b} of [a,b],


The total variationMathworldPlanetmath Vγ of γ is defined by

Vγ=sup{v(γ,P):P is a partition of [a,b]}.

It can be shown that, if X is either or , every continuously differentiable (or piecewise continuously differentiable) function γ:[a,b]X is of bounded variation (, and


Also, if γ is of bounded variation and f:[a,b]X is continuousMathworldPlanetmath, then the Riemann-Stieltjes integral abf𝑑γ is finite.

If γ is also continuous, it is said to be a rectifiable path, and V(γ) is the length of its trace.

If X=, it can be shown that γ 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