Convex continuous functions

Question

\begin{equation} w \left (\dfrac{x+y}{2} \right ) \le \dfrac{1}{2}(w(x) +w(y)) \quad \mbox{for all} \quad x,y \in \Omega, \end{equation} is a sufficient condition to a continuous function $w \in C^0(\Omega)$ be convex?

Answer 1

Yes---as long as $\Omega$, the domain of the function $w$, is a convex set.

This is called the "midpoint test" for convexity. I did a little Googling and found this post that offers a proof for the scalar case. And since a function is convex if and only if it is convex along every line, that should be sufficient.

EDIT: Actually, it looks like this is a duplicate question.