Will continuous extension preserve strict convexity?

Question

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.

Answer 1

I believe that $f(x,y) = \dfrac{x^2}{\sqrt{1+y}}$, considered on $[0,1]^2$, is a counterexample of the type you describe.

Answer 2

You may also take $f(x,y) =e^{x^2}y^2$ for $x\neq 0$.