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I just want to prove that a function from $\mathbb{R}$ to $\mathbb{R}$ is Lipschitz continuous, if and only if $\exists\, g\in L^p(\mathbb{R}) $ such that $\forall\, x, \, y $, $$f(y)-f(x)\le \int_x^yg(t) ~dt.$$ Setting $M=||g||_\infty$, I can prove that the right side implies the left, but I do not know how to prove the direction from left to right.

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Why would $g$ be bounded? – Davide Giraudo Nov 30 '12 at 20:13

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