# Hint for real analysis question [duplicate]

Suppose that $f$ is one-to-one and continuous on [$a,b$]. Prove that $f$ is either strictly increasing or strictly decreasing on [$a,b$].

• Suppose it weren't. Use the intermediate value theorem to derive a contradiction. – Daniel Fischer Oct 23 '13 at 22:30

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 $f$.
• Most (in the sense that the complement is of the first category) elements of $C^0([a,b])$ are nowhere differentiable. – Daniel Fischer Oct 23 '13 at 22:27
Hint: Go by contradiction. If its not monotone then we have some $p \le q \le r$ such that
$$f(p) \ge f(q) \le f(r)$$