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f(f (x)) = f (x)^2, f (1) = 1, continuous monotonic

Source: Austrian Polish 1986 APMC

April 30, 2020
continuousfunctionalfunctional equationcompositionalgebra

Problem Statement

Find all continuous monotonic functions f:R→Rf : R \to R that satisfy f(1)=1f (1) = 1 and f(f(x))=f(x)2f(f (x)) = f (x)^2 for all x∈Rx \in R.
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