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f(x)=x has real solution(s) if f(f(x))=x has real solution(s), f continuous

Source: 1985 Greece MO Grade XII p2

September 6, 2024

algebracontinuous function

Problem Statement

Conside the continuous

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

. It is also know that equation

f(f(f(x)))=x

has solution in

\mathbb{R}

. Prove that equation

f(x)=x

has solution in

\mathbb{R}