Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function that satisfies the conditions: $ \text{(i)}\quad \lim_{x\to\infty} (f\circ f) (x) =\infty =-\lim_{x\to -\infty} (f\circ f) (x) $ $ \text{(ii)}\quad 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?
Problem
Source: Romanian National Olympiad 2000, Grade XI, Problem 4
Tags: function, real analysis, Darboux, limits, epsilon-delta, second iterate