Bounded variation

Suppose f is a real-valued function on the interval [a, b] on the real line. The total variation of f on that interval is

the supremum running over all partitions { 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.

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Bounded variation