Bounded variation
Suppose
f is a
realvalued 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 {
x_{1}, ...,
x_{n} } of the interval [
a,
b]. In effect, the total variation is the vertical component of the arclength 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 RiemannStieltjes integrals.
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