Real Analysis/Total Variation

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Real Analysis
Total Variation

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 xk such that x0=a, xk>xk-1, and such that xn=b. The total variation t of a function on the interval [a,b] is the supremum

t= sup{a|a= \sum_{k=1}^{n}\|f(x_k)-f(x_{k-1})\|} and xk is a partition of [a,b]}.

If this supremum exists, then 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.