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$f, g\colon \mathbb {R}\to \mathbb {R}$ continuous functions such that $f(g(x))=x$. Prove that $g(f(x))=x$.
Since $f$ is surjective, we basically have to prove that $f$ is injective. For $g$ is clearly, but how can I prove that $f$ is injective ? Please, help me!

N. F. Taussig
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Lucas McAllister
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    This holds even when $f$ is not surjective. Hint: a continuous function is invertible if and only if it is strictly increasing or strictly decreasing. You will easily be able to google a reference to this . – Kurt G. Mar 30 '22 at 11:57
  • $f$ is surjective because $f(g(x))=x$. It was just an observation. And I know that stuff already. I saw that $g$ is a strictly monotonic function, but I haven't thought that it will help. But now that I think about it, it might help. – Lucas McAllister Mar 30 '22 at 12:06
  • How about $g(x)=\tan(x)$ and $f(x)=\arctan(x)$ ? That ain't surjective, at least not in $\mathbb R$. – Kurt G. Mar 30 '22 at 12:14
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    @KurtG. $\tan(x)$ is not continuous on ${\mathbb R}$. – Gribouillis Mar 30 '22 at 12:15
  • @Gribouillis . Right ! OP wants $g$ to be defined on the whole $\mathbb R$. Take it back. – Kurt G. Mar 30 '22 at 12:16

1 Answers1

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Hint:

  • As $g$ is continuous and injective, it is strictly monotonic
  • if $\ell := \lim_{x \to \infty} g(x) \in {\mathbb R}$, then $\lim_{x\to\infty} f(g(x)) = f(\ell)$, contradicting $f(g(x)) = x$. Hence $\lim_{x\to\infty} g(x) = \pm \infty$
  • Same result with $\lim_{x\to -\infty}g(x)$
  • conclude that $g$ is surjective.
  • If $y=g(x)$ then $f(y)= f(g(x)) = x$, hence $g(f(y)) = g(x) = y$.
Gribouillis
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