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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](http://en.wikipedia.org/wiki/Lipschitz_continuity). On a sunny day, [mollification](http://en.wikipedia.org/wiki/Mollifier) will work, but in the absence of global Lipschitzness or the presence of boundaries, you may have to work a bit harder.2012-05-01

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