Why is every such function constant: $f(x)=f(x^2)$ for $x \in [0,1]$? Let $f:[0,1] \to \mathbb{R}$ be a continuous function such that $f(x)=f(x^2)$ for all $x \in [0,1]$.
Any hint/idea for proving that $f$ has to be constant?

Note that $f(x) = f(x^2) = f(x^4) = \cdots = f(x^{2^n})$.
If $x \in [0,1)$, we see that $x^{2^n} \to 0$, and by continuity we have $f(x^{2^n}) \to f(0)$, from which we get that $f(x) = f(0)$.
By continuity we also get that $f(1) = \lim_{x \to 1} f(x) = f(0)$.
