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.

Proof
As $f$ is midpoint-concave, then $-f$ is midpoint-convex.

From Continuous Midpoint-Convex Function is Convex, $-f$ is convex.

From Real Function is Concave iff its Negative is Convex, $f$ 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