Continuous Midpoint-Convex Function is Convex

Theorem
Let $f$ be a real function which is defined on a real interval $I$.

Let $f$ be midpoint-convex and continuous on $I$.

Then $f$ is convex.

Also see

 * Continuous Strictly Midpoint-Convex Function is Strictly Convex


 * Continuous Midpoint-Concave Function is Concave
 * Continuous Strictly Midpoint-Concave Function is Strictly Concave