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Let $f$ be a convex function. How to prove that there is a sequence of differentiable convex function that converges uniformly to $f$?

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Depends on the domain of $f$, and if $f$ is globally Lipschitz. On a sunny day, mollification will work, but in the absence of global Lipschitzness or the presence of boundaries, you may have to work a bit harder. –  Harald Hanche-Olsen May 1 '12 at 10:47

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