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f^n(x) = 1 implies f(1) = 1

Source: Poland Math Olympiad 1993 Round 1 #6

June 3, 2023
function

Problem Statement

The function f:R⟶Rf: R \longrightarrow R is continuous. Prove that if for every real number xx, there exists a positive integer nn, such that (f∘f∘...∘f)⏟n(x)=1\underbrace{(f \circ f \circ ... \circ f)}_{n}(x) = 1, then f(1)=1f(1) = 1.
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