IMC 2006 · Problem 1

Let $f : \mathbb{R} \to \mathbb{R}$ be a real function. Prove or disprove each of the following statements.

(a) If $f$ is continuous and $\operatorname{range}(f) = \mathbb{R}$ then $f$ is monotonic.

(b) If $f$ is monotonic and $\operatorname{range}(f) = \mathbb{R}$ then $f$ is continuous.

(c) If $f$ is monotonic and $f$ is continuous then $\operatorname{range}(f) = \mathbb{R}$.