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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 $\displaystyle f(x)+\lim_{n\to\infty}\left((-1)^n\sum_0^{n-1}(-1)^{k+1}x^{2^k}\right)=0$

Gabriel Romon
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  • $f(0)=0$ is fine but i do not understand the other equation... –  Jun 25 '14 at 09:17
  • @PraphullaKoushik there may be a mistake, but I just iterated the identity on $f(x^{2^n})$ and took the limit at $n \to \infty$ – Gabriel Romon Jun 25 '14 at 09:20

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