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Since
the Jacobian matrix is invertible: let
be its inverse. Choose and such that
Let
and consider the mapping
If we have
Let us verify that is a contraction mapping. Given
, by the Mean-value Theorem on
we have
Also notice that
. In fact, given ,
So
is a contraction mapping and hence by the contraction principle there exists one and only one solution to the equation
i.e. is the only point in such that .
Hence given any
we can find which solves . Let us call
the mapping which gives this solution, i.e.
Let
and . Clearly
is one to one and the inverse of is . We have to prove that is a neighbourhood of . However since is continuous in we know that there exists a ball
such that
and hence we have
.
We now want to study the differentiability of . Let be any point, take
and
so small that
. Let and define
.
First of all notice that being
we have
and hence
On the other hand we know that is differentiable in that is we know that for all it holds
with
. So we get
So
that is
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