Continuous Strictly Midpoint-Convex Function is Strictly Convex

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

Then $f$ is strictly convex.

Also see

 * Continuous Midpoint-Convex Function is Convex


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