Bounded variation

In mathematics, given f, a real-valued function on the interval [a, b] on the real line, the total variation of f on that interval is

<math>\mathrm{sup}_P \sum_i | f(x_{i+1})-f(x_i) | <math>

the supremum running over all partitions P = { x₁, ..., x_n } of the interval [a, b]. In effect, the total variation is the vertical component of the arc-length of the graph of f. The function f is said to be of bounded variation precisely if the total variation of f is finite.

Functions of bounded variation are precisely those with respect to which one may find Riemann-Stieltjes integrals. There is another characterisation available in distribution theory; they are the functions whose derivative in the distributional sense is a measure.

Another characterization states that the functions of bounded variation are exactly those f which can be written as a difference g-h, where both g and h are monotone.