Midpoint convex and monotonicfunction

Question

If a function is monotonic and midpoint convex, then is it convex or continous? Is there a known result of this sort?

Answer 1

Obviously, you meant monotonically increasing. By iteration mid-point convexity gives $f(ax+(1-a)y) \leq af(x)+(1-a)f(y)$ whenever $a=\frac m {2^{n}}$ for some positive integers $n,m$ and $x<y$. Given any $a \in (0,1)$ we can find a sequence $\{a_k\}$ of numbers of above type with $a_k <a$ conveging to $a$. Now $$f(ax+(1-a)y) =f((y+a(x-y))$$ $$\leq f((y+a_k(x-y)) =f(a_kx+(1-a_k)y) \leq a_kf(x)+(1-a_k)f(y)$$ Now let $k \to \infty$ to see that $f$ is convex. Of course, convex functions on open intervals are necessarily contin uous.