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Let f(x) be a continuous function on an interval [a,b]. A partition of f(x) on the interval [a,b] is a sequence x_{k} such that x_{0}=a, x_{k}>x_{k-1}, and such that x_{n}=b. The total variation t of a function on the interval [a,b] is the supremum

t= sup{a|a=} and x_{k} is a partition of [a,b]}.

If this supremum exists, the the function is of bounded variation on [a,b]. If a real function is of bounded variation over its whole domain, then it is called a function of bounded variation.