RO-NMO 2000 Problem 114

Let $f:\mathbb{R}\longrightarrow\mathbb{R}$ be a function that satisfies the conditions: $\text{(i)} \lim_{x\to\infty} (f\circ f) (x) =\infty =-\lim_{x\to -\infty} (f\circ f) (x)$ $\text{(ii)} f$ has Darboux's property

a) Prove that the limits of $f$ at $\pm\infty$ exist. b) Is possible for the limits from a) to be finite?