Continuous Midpoint-Concave Function is Concave

Theorem
Let $f$ be a real function which is defined on a real interval $I$. Let $f$ be midpoint-concave and continuous on $I$.

Then $f$ is concave.

Also see

 * Continuous Strictly Midpoint-Concave Function is Strictly Concave


 * Continuous Midpoint-Convex Function is Convex
 * Continuous Strictly Midpoint-Convex Function is Strictly Convex