Differentiable Bounded Convex Real Function is Constant

Theorem
Let $f$ be a real function which is:
 * $(1): \quad$ differentiable on $\R$
 * $(2): \quad$ bounded on $\R$
 * $(3): \quad$ convex on $\R$.

Then $f$ is constant.

Proof
Let $f$ be differentiable and bounded on $\R$.

Let $f$ be convex on $\R$.

Let $\xi \in \R$.

$\map {f'} \xi > 0$.

Then by Mean Value of Convex Real Function it follows that:
 * $\map f x \ge \map f \xi + \map {f'} \xi \paren {x - \xi} \to + \infty$ as $x \to +\infty$

and therefore is not bounded.

Similarly, suppose $\map {f'} \xi < 0$.

Then by Mean Value of Convex Real Function it follows that:
 * $\map f x \ge \map f \xi + \map {f'} \xi \paren {x - \xi} \to + \infty$ as $x \to -\infty$

and therefore is likewise not bounded.

Hence $\map {f'} \xi = 0$.

From Zero Derivative implies Constant Function, it follows that $f$ is constant.

Also see

 * Differentiable Bounded Concave Real Function is Constant