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Is there any example of a function $f(x)$ differentiable at $x=0$, with an inverse function that is not continuous when $x=0$? Any help where to start, or maybe even if someone has the example of such a function would be greatly appreciated.

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    So you wnat $f,g$ such that $f(0)=g(0)=0$, $f(g(x))=g(f(x))=x$, $f$ differentiable at $0$, $g$ not continuous at $0$?2012-12-10

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