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Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be continuously differentiable. Suppose there is a function $g:\mathbb{R}^n\to\mathbb{R}^n$ such that $fg=gf=1_{\mathbb{R}^n}$.

Does it follow that $g$ is differentiable?

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    @NickStrehlke Yes, indeed you're right, I take back what I said in my previous comment.2012-12-12

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