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.

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

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

$\blacksquare$

Also see

• Continuous Strictly Midpoint-Concave Function is Strictly Concave

• Continuous Midpoint-Convex Function is Convex
• Continuous Strictly Midpoint-Convex Function is Strictly Convex