condition of convexity when midpoint convex

Question

Let $f:I \rightarrow R$ be a function satisfying the equation $f(\dfrac{x+y}{2}) \leq \dfrac{f(x)+f(y)}{2}$

The question is,

1)Is $f$ continuous when $I$ is closed?
2)Is $f$ continuous when $I$ is opened?

While searching the internet, there was proof about convexity and continuity, but I can't find about the relation of midpoint convexity and continuous.

Also, is there any intuitive difference about the question (1) and (2)? (I've found some counterexamples about continuity and counterexamples, but none of them showed the proceedings to reach about the counterexample.

Answer 1

A standard (counter)-example are $\mathbb Q$-linear functions on $\mathbb R$. They satisfy the midpoint-convexity inequality but not all of them are continuous.