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Found problems: 1

2000 Romania National Olympiad, 4
Tags: real analysis , function , limit , darboux , epsilon-delta , second iterate
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 [b]a)[/b] Prove that the limits of $ f $ at $ \pm\infty $ exist. [b]b)[/b] Is possible for the limits from [b]a)[/b] to be finite?