# Is a convex function defined on a convex open subset of $\mathbb R^n$ continuous?

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?

-
Can you prove it for $n = 1$? See here for that case. – t.b. Jul 18 '11 at 9:09
@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. – Jeff Oct 20 '12 at 19:26
I 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. – Jeff Oct 20 '12 at 19:34
@Jeff: See e.g. Theorem 3.3.1 in these notes – t.b. Nov 29 '12 at 14:54

## 1 Answer

Here is a link which I found using Google. See the proof of $\textbf{Theorem 3.3.1.}$

-
Indeed an illuminating proof! thanks. – cheng Jul 19 '11 at 1:40
@cheng: then accept this as an answer. – user9413 Jul 19 '11 at 3:30