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Do there exist continuous functions $f,g: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(g(x))=x^2$ and $g(f(x))=x^3$ for all $x \in \mathbb{R}$?

My attempt: Since $x^3$ is a bijection, we have $f$ is injective and $g$ is surjective. Then I don't know how to proceed from here.

My feeling tells me that there doesn't exist such function. But I don't know how to show it.

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  • $\begingroup$ If you look at $f(g(f(x)))$ and evaluate it in two different ways, you get $f(x)^2 = f(x^3)$. Likewise, if you evaluate $g(f(g(x)))$ in two different ways, you get $g(x)^3 = g(x^2)$. I'm not sure if this is helpful or not. $\endgroup$ Commented Jun 3, 2015 at 4:05

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From $f(g(x)) = x^2$ and the fact that $g$ is surjective, you get that the image of $f$ is $[0,+\infty)$. But this is impossible for a continuous injection from $\mathbb{R}$ to $\mathbb{R}$ (which must be monotonic).

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