Let $E$ be a locally convex TVS, $C$ a convex subset of $E$ and $g:E\to]-\infty,\infty]$ a convex function. I was trying to prove the following
If $g$ is finite on $C$, then $g$ is finite on $\overline C$.
but I think this is probably wrong, if $C\neq \overline C$ and we set $$g(x)=\begin{cases}0&\text{if } x\in C \\ +\infty&\text{otherwise}\end{cases}$$ then I think this function is convex indeed convexity can be broken only if $g(x)=g(y)=0$ and there is $\lambda\in]0,1[$ such that $g(\lambda x+(1-\lambda) y)=\infty$, but since $\lambda x+(1-\lambda) y\in C$ by convexity, we do get $g(\lambda x+(1-\lambda) y)=0$.
Does that seem correct ?