Find all continuous $f :[0,1]\to \mathbb R$ such that $\forall x\in [0,1], f(x)+f(x^2)=x$
I suspect there are none.
I made little progress so far, but it's worth noticing that $f(0)=0$ and $\displaystyle f(x)+\lim_{n\to\infty}\left((-1)^n\sum_0^{n-1}(-1)^{k+1}x^{2^k}\right)=0$