If $f(g(x)) = x$ for all $x$ and $f$ and $g$ are continuous. Does it necessary follow that $g = f^{-1}$? Or do we need $g(f(x)) = x$ as well?
Asked
Active
Viewed 38 times
1 Answers
1
If "all $x$" means all real $x$, and "continuous" means "continuous on $\mathbb R$", then yes. It's clear that $g$ is one-to-one. $g({\mathbb R})$ is then an open interval. Suppose that interval is not $\mathbb R$. Consider e.g. the case where $g(x)$ increases to a finite value $b$ as $x \to +\infty$. Then $\lim_{y \to b-} f(y) = \lim_{x \to +\infty} f(g(x)) = \infty$, contradicting continuity of $f$ at $b$.
Robert Israel
- 448,999
-
@sidth:What are the domains and ranges of $f$ and $g$? Are they defined on arbitrary sets $X,Y$ or are they defined on $\mathbb{R}$? – user64066 Jul 19 '13 at 06:09
-
@user64066, based on your comment there appears to be two cases. Could you show me both? – Lemon Jul 19 '13 at 19:15
-
@Robert, your explaination is excellent. I understand it (only because I have some background in analysis). Is there a more fundamental explanation to someone of pre-calculus? – Lemon Jul 19 '13 at 19:26