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 $x \in \mathbb{R}$?

Question

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.

Answer 1

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).