Is it true that for F : R n → R m which has an inverse function F − 1 : R m → R n , says F is diﬀerentiable at a ∈ R n and F − 1 is diﬀerentiable at b = F ( a ) ∈ R m ，then m must be equal to n?

Question

Is it true that for   F  :            R        n    →            R        m   which has an inverse function       F          −      1        :            R        m    →            R        n  , says   F is diﬀerentiable at   a  ∈            R        n   and       F          −      1       is diﬀerentiable at   b  =  F  (  a  )  ∈            R        m  ，then   m must be equal to   n?

anclarlo5h12v · Accepted Answer

The Inverse Function Theorem requires a continuously differentiable function.Your assumption is true. The Invariance of Domain Theorem says that   F cannot be bijective and continuous if   m  ≠  n.

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