Real Function both Convex and Concave is Linear

Theorem

Let $f$ be a real function which is both convex and concave.

Then $f$ is a linear function.

Proof

Let $f$ be both convex and concave on a subset $S \subseteq \R$ of the real numbers $\R$.

Then by definition:

\(\ds \forall x, y \in S: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \, \)

\(\ds \map f {\alpha x + \beta y}\)

\(\le\)

\(\ds \alpha \map f x + \beta \map f y\)

Definition of Convex Real Function

\(\ds \forall x, y \in S: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \, \)

\(\ds \map f {\alpha x + \beta y}\)

\(\ge\)

\(\ds \alpha \map f x + \beta \map f y\)

Definition of Concave Real Function

\(\ds \leadsto \ \ \)

\(\ds \map f {\alpha x + \beta y}\)

\(=\)

\(\ds \alpha \map f x + \beta \map f y\)

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Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): convex function