What does the inverse function say when det f ′ ( x ) doesn't equal 0?

Question

What does the inverse function say when   det      f    ′    (  x  ) doesn&#039;t equal   0?

alomjabpdl0 · Accepted Answer

I&#039;m assuming you&#039;re talking about the inverse function theorem. Basically, if you have a function   f  :  U  →  V where   U and   V are open sets in Euclidean space, and furthermore, if   d  e  t  (  D  f  (  p  )  ) where   p  ∈  U is non-zero, then there is an inverse differentiable function from a neighbourhood of   f  (  p  ) to a neighbourhood of   p. Furthermore, the derivative of       f          −      1       at   f  (  p  ) is   (  D  f  (  p  )      )          −      1      . Intuitively, this says that   f is locally invertible (or more interestingly, a local differentiable homeomorphism).

What does the inverse function say when <mo

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