Let $K$ be a convex open set in $\mathbb R^n$ and $f$ a convex function defined on $K$; how to show that $f$ is continuous?
Is a convex function defined on a convex open subset of $\mathbb R^n$ continuous?
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real-analysis
convex-analysis
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5Can you prove it for $n = 1$? See [here](http://math.stackexchange.com/questions/24676/convex-function-in-open-interval-is-continuous) for that case. – 2011-07-18
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0@t.b. can you say more about why n=1 generalizes to the present case? The link in the only answer here is broken. It seems like all you get is cross-sectional Lipschitz and continuity properties. – 2012-10-20
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0I should mention that I heaved a sigh of relief when I thought that "cross-sectional Lipschitz" would be enough, but the thing is it's not just one lipschitz constant, or even one for each of the n axial directions. It's a different constant for every cross section. – 2012-10-20
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0@Jeff: See e.g. Theorem 3.3.1 in [these notes](http://ljk.imag.fr/membres/Anatoli.Iouditski/cours/convex/chapitre_3.pdf) – 2012-11-29