Real Analysis/Total Variation

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Real Analysis Total Variation

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Let f be a continuous function on an interval [a,b]. A partition of f on the interval [a,b] is a sequence x_k such that a=x₀< x₁ <...< x_k-1 < x_k < ...x_n=b. The total variation t of a function on the interval [a,b] is the supremum

t= sup{$\sum _{k=1}^{n}|f(x_{k})-f(x_{k-1})|$ : x_k 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.