Total variation - Knowlpedia

The total variation of a function $\alpha:[a,b]\to\mathbb R$ is the extended nonnegative number

V_a^b(\alpha)=\sup_P \sum_{i=1}^n |\alpha(x_i)-\alpha(x_{i-1})|,

where the supremum is taken over all partitions $P=\{a=x_0<\cdots<x_n=b\}$ .

Remarks

Total variation measures how much $\alpha$ accumulates change along the interval and is the defining quantity for a function of bounded variation. For monotone functions it reduces to the net change.

Examples

If $\alpha$ is increasing on $[a,b]$ , then $V_a^b(\alpha)=\alpha(b)-\alpha(a)$ .
For $\alpha(x)=\sin x$ on $[0,2\pi]$ , one has $V_0^{2\pi}(\alpha)=4$ .