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Romania National Olympiad 2011 - Grade XI - problem 3

Source:

April 19, 2011

functionreal analysisreal analysis unsolved

Problem Statement

Let

g:\mathbb{R}\to\mathbb{R}

be a continuous and strictly decreasing function with

g(\mathbb{R})=(-\infty,0)

. Prove that there are no continuous functions

f:\mathbb{R}\to\mathbb{R}

with the property that there exists a natural number

k\ge 2

so that :

\underbrace{f\circ f\circ\ldots\circ f}_{k\text{ times}}=g