Question 173
Find all continuous functions $f: [0,1] \to [0,1]$ satisfying the following condition: For any $x \in [0,1]$, there exists a positive integer $n$ such that $f^n(x) = x$.
(For $x \in [0,1]$, $f^n(x)$ is defined through $f^1(x) = f(x)$ and $f^{k+1}(x) = f(f^k(x))$ for all positive integers $k$.)
(For $x \in [0,1]$, $f^n(x)$ is defined through $f^1(x) = f(x)$ and $f^{k+1}(x) = f(f^k(x))$ for all positive integers $k$.)