Mid-point convexity does not imply convexity

Question

A function $f: X \rightarrow \mathbb{R}$ is said to be mid-point convex if for all $x, y \in X$, we have $$f(\frac{x + y}{2}) \leq \frac{f(x) + f(y)}{2}. $$ Can you please give an example of a function which is mid-point convex but not convex?

Answer 1

That is a theorem on convex analysis but it is stated for continuous function. For making counterexample you can remove the continuity from the condition such as $f(x)=x^2$ for $x\in \mathbb{Q}$ and $0$ otherwise. I think that, it is a counter example.