Convex function in its interior Let $f$ be a convex function on an open subset of $R^{n}$. How to prove $f$ is continuous in the interior of its domain.  
For $n=1$, let $f$ be convex on the set $(a,b)$ with $a<s<t<u<b$  
Then using the inequality $\frac{f(t)-f(s)}{t-s} \leq \frac{f(u)-f(s)}{u-s} \leq \frac{f(u)-f(t)}{u-t} $
We can prove it for $n=1$. 

You can see the solution in the following lecture note
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