Let $f$ be a convex function. How to prove that there is a sequence of differentiable convex function that converges uniformly to $f$?
A sequence of differentiable convex function converges uniformly to a convex function?
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functions
convex-analysis
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1Depends 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