Differentiable Bounded Convex Real Function is Constant

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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 $\xi \in \R$.

Aiming for a contradiction, suppose $f' \left({\xi}\right) > 0$.

Then by Mean Value of Convex Real Function it follows that:

$f \left({x}\right) \ge f \left({\xi}\right) + f' \left({\xi}\right) \left({x - \xi}\right) \to + \infty$ as $x \to +\infty$

and therefore is not bounded.

Similarly, suppose $f' \left({\xi}\right) < 0$.

Then by Mean Value of Convex Real Function it follows that:

$f \left({x}\right) \ge f \left({\xi}\right) + f' \left({\xi}\right) \left({x - \xi}\right) \to + \infty$ as $x \to -\infty$

and therefore is likewise not bounded.

Hence $f' \left({\xi}\right) = 0$.

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

$\blacksquare$


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