Hi, everyone:(adsbygoogle = window.adsbygoogle || []).push({});

I was wondering if anyone knew of any extensions to the inverse function theorem

to this effect:

If f is a differentiable map, and df(x) is non-zero, then the IFT guarantees

there is a nhood (neighborhood) U_x containing x , such that [itex] f|(U_x) [/itex]

a diffeom. [itex] U_x-->f(U_x) [/itex].

Now, under what conditions on f , can we be guaranteed to have that f

has a global differentiable inverse?, i.e, f has a global inverse [itex] f^-1 [/itex] and

[itex] f^-1 [/itex] is differentiable . I imagine df(x) not 0 for all x is necessary, but not

sure that it is sufficient.

For simple cases like f(x)=[itex] x^2 [/itex] this is true in any interval [a,b]

not containing zero, and it may be relatively easy to do a proof of the claim

above for maps f:IR->IR . But I have no idea how well this would generalize

to maps f:[itex] R^n\to\mathbb{R^n} [/itex]

Any Ideas?

Thanks.

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# Generalizations of the Inv. Function Thm.

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