Midconvex functions that are not convex are not Lebesgue measurable
A function f : I ⊆ R → R is midpoint convex
if
f(a/2 + b/2)≤f (a ) /2+ f (b )/2
for all a, b∈ I.
There is a discussion at math.stackexchange which is relevant to your question. This is the direct link.
http://math.stackexchange.com/questions/71019/example-of-a-function-such-that-varphi-left-fracxy2-right-leq-frac-va
You should pay particular attention to the answer offered by Robert Israel, who explains why an explicit example can not be constructed.
For continuous functions, "midpoint convex" is equivalent to "convex" as Mikkelson's link already noted. This is very similar to the argument that an additive continuous function has the form f(x)=Ax with discontinuous counterexamples easily constructed in terms of a Hamel base for the reals as a vector space over the rational field. Indeed, any such an example is also an example here: midpoint convex (with equality!) but not convex.
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