Theorem: Every bounded differentiable function $f\colon \mathbb{R}\to \mathbb{R}$ is constant.
Proof.
By assumption there exist real numbers $M,N$ such that
$$N\leq f(x) \leq M.$$
Taking derivatives we get
$$0\leq f'(x)\leq 0.$$
Hence $f'(x)=0$ so $f$ is constant. QED
