Total variation

The supremum of sums of absolute increments over all partitions.

Total variation

A total variation of a function $\alpha:[a,b]\to\mathbb R$ on $[a,b]$ is the 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 $P=\{x_0,\dots,x_n\}$ of $[a,b]$ .

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 $V_0^{2\pi}(\alpha)=4$ .