Suppose that $f$ is one-to-one and continuous on [$a,b$]. Prove that $f$ is either strictly increasing or strictly decreasing on [$a,b$].
Hint Use the intermediate-value theorem to get a contradiction if you assume $$\exists x,y \in [a,b] : f'(x) < 0 \wedge f'(y) > 0$$ Use Weierstraß to acqurie a differentiable function with norm arbitrarily close to $f$.
Hint: Go by contradiction. If its not monotone then we have some $p \le q \le r$ such that
$$f(p) \ge f(q) \le f(r)$$
Then apply the intermediate value theorem to get a contradiction.