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Working on Hilbert space, is it possible to find a real valued convex lower semicontinuous function in which it does not continuous at any point in a space. If not, how can we prove it?

user428714
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  • No. Note that a convex function is locally Lipshitz, see, eg, here – Thomas May 10 '17 at 17:35
  • Convexity implies continuity at every point except possibly the end points, so this is not possible. – Nigel Overmars May 10 '17 at 17:35
  • Sorry, I should discuss with a Hilbert space, is it possible? – user428714 May 10 '17 at 17:36
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    @Thomas, convex $\textbf{continous}$ functions in a normed space are locally Lipschitz, but not every convex function is continuous in a normed space, so this question is legit – John D May 27 '18 at 09:39
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    @Magnusseen yes, you are right. Thanks for pointing that out. The answer to the question can still be given and can be found, e.g., here: http://www.mat.unimi.it/users/libor/AnConvessa/continuity.pdf – Thomas May 27 '18 at 10:55

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