I'm trying to show that a continuous function $f:(a,b) \to \mathbb{R}$ with the mid-point convexity property is convex.
Midpoint convexity is $f((x+y)/2) \leq \frac{1}{2}f(x) + \frac{1}{2} f(y)$.
Convexity is $f(tx + (1-t)y) \leq tf(x) + (1-t)f(y)$ for all $t \in [0,1]$.
Starting with $f((x_1 + x_2)/2) \leq \frac{1}{2} f(x_1) + \frac{1}{2} f(x_2)$, I could show
$$f((x_1 + x_2+ x_3 + x_4)/4) \leq \frac{1}{4} f(x_1) + \frac{1}{4} f(x_2)+ \frac{1}{4} f(x_3)+ \frac{1}{4} f(x_4)$$
Using induction I get for all $n$
$$f\left(2^{-n}\sum_{j=1}^{2^n}x_j\right) \leq 2^{-n}\sum_{j=1}^{2^n}f(x_j). $$
I can see that continuity lets me take a limit inside $f()$, but I'm not sure how to continue to prove the general property.
