If a function $f$ satisfies $$ (1) \; \; \; f({x+y\over2})\le {f(x)+f(y)\over2}, $$and it is continuous, then it is not hard to see that $f$ is indeed convex. On the other hand, a discontinuous additive function, which exists by the axiom of choice, satisfies (1) but is not convex.

My question is, can we find functions satisfying (1) which are not convex, without using the axiom of choice? Or, are there models of set theory in which all functions satisfying (1) are convex (continuous)?