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High Quality Content by WIKIPEDIA articles! In mathematics, the total variation of a real-valued function ?, defined on an interval [a, b] ? R is a measure of the one-dimensional arclength of the curve with parametric equation x ? ?(x), for x ? [a,b]. he total variation of a continuously differentiable function can be given as the integral V^a_b(f) = int _a^b f'(x) , dx. The total variation of an arbitrary real valued function ? defined on [a,b] is given by the more general formula V^a_b(f)=sup_P sum_{i=0}^{n_P-1} f(x_{i+1})-f(x_i) , , where the supremum runs over the set of all partitions P of the given interval.