Function of bounded variation

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

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

Functions of bounded variation are important because they provide a broad class of integrator functions for the Riemann–Stieltjes integral and admit structural decompositions such as the Jordan decomposition .

Examples:

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