Real Function with Negative Derivative is Decreasing

Theorem
Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.

If $\forall x \in \openint a b: \map {f'} x \le 0$, then $f$ is decreasing on $\closedint a b$.

Proof
Let $c, d \in \closedint a b$ such that $c < d$.

Then $f$ satisfies the conditions of the Mean Value Theorem on $\closedint c d$.

Hence:
 * $\exists \xi \in \openint c d: \map {f'} \xi = \dfrac {\map f d - \map f c} {d - c}$

Let $f$ be such that:
 * $\forall x \in \openint a b: \map {f'} x \le 0$

Then:
 * $\map {f'} \xi \le 0$

and hence:
 * $\map f d \le \map f c$

Thus $f$ is decreasing on $\closedint a b$.