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A function is called absolutely continuous on an interval I if for any ε > 0 there exists a δ > 0 such that for any pair-wise disjoint subintervals (xk, yk) = k = 1, 2, . . . , n, of I such that ∑(xk - yk) < δ we have ∑|f(xk) - f(yk)| < e. Show that if f satisfies a Lipschitz condition on I, then f is absolutely continuous on I. |
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