How to prove that $f$ is convex function if $f(\frac{x+y}2)\leq \frac12f(x) + \frac12f(y)$ and $f$ is continuous?

Question

Let $f:(a,b) \mapsto \mathbb{R}$ be continuous function such that $f(\frac{x+y}{2})\leq \frac{1}{2}f(x) + \frac{1}{2}f(y) \;\; \forall x,y \in (a,b)$

Show that $f$ is convex function. Please give me hints to prove it.

Answer 1

Hint: Use the above condition to show that $f(\lambda x+(1-\lambda)y)\leq \lambda f(x)+(1-\lambda)f(y)$ where $\lambda$ is a dyadic rational.

Then show that the dyadic rationals are dense in $\mathbb{R}$; since limits preserve inequalities, you can then take a limit to any real number, and you're done.