lean_workbook_plus_19946

Find the function $f(x)$ that satisfies the given conditions: $f: \mathbb{R} \to \mathbb{R}$ is continuous, $f(0) = 0$, $f(1) = 1$, and $f_{(n)}(x) = x$ for all $x \in [0, 1]$ and $n \in \mathbb{N}^*$, where $f_{(n)}(x)$ represents the $n$-th iterate of $f(x)$.