The set of functions $P$ for which $P(x)^2=P(x^2)$ for all $x\in \mathbb{R}$

Question

We showed that all of polynomials satisfy the equation $$P(x)^2 = P(x^2)$$ are $P(x) = x^n, n \in \mathbb{N}$.

If $P(x)$ is only a continuous function, can we describe the set of solutions of the above equation, i.e., $P(x)^2 =P(x^2)$?