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Turkey NMO 2000 Problem 6, Find All Functions Such That ...

Source: Turkey NMO 2000 Problem 6

September 29, 2011
functioninequalitiesalgebra proposedalgebra

Problem Statement

Find all continuous functions f:[0,1]→[0,1]f:[0,1]\to [0,1] for which there exists a positive integer nn such that fn(x)=xf^{n}(x)=x for x∈[0,1]x \in [0,1] where f0(x)=xf^{0} (x)=x and fk+1=f(fk(x))f^{k+1}=f(f^{k}(x)) for every positive integer kk.
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