Continuous Midpoint-Concave Function is Concave

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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.

$\blacksquare$


Also see