The problem I am thinking about is like follows.
Suppose that $h$ is a strictly convex function on an open convex set $S$. Then, we extend $h$ continuously to the closure of $S$ that is denoted by $\bar{S}$. Let $\bar{h}$ be the continuous extension of $h$ on $\bar{S}$. Is $\bar{h}$ strictly convex on $\bar{S}$?
I think this might not be true but I have not found a counterexample. I guess there might be a function that is strictly convex on an open square, but is only convex on one of its sides.
Thanks for help.