Knowlpedia

Function of bounded variation

A function whose total variation on an interval is finite.

A function of bounded variation on [a,b][a,b] is a function α:[a,b]→R\alpha:[a,b]\to\mathbb R whose on [a,b][a,b] is finite, i.e.

Vab(α)=sup⁡P∑i=1n∣α(xi)−α(xi−1)∣<∞,V_a^b(\alpha)=\sup_P \sum_{i=1}^n |\alpha(x_i)-\alpha(x_{i-1})|<\infty,

where the supremum is taken over all P={x0,…,xn}P=\{x_0,\dots,x_n\} of [a,b][a,b].

Applications

Functions of bounded variation are important because they provide a broad class of for the and admit structural decompositions such as the .

Examples
  • Any on [a,b][a,b] has bounded variation.
  • The function α(x)=sin⁡x\alpha(x)=\sin x has bounded variation on [0,2π][0,2\pi].