The inverse function theorem states that, given a smooth function 
,
if the Jacobian is invertible at 0, there is a neighborhood 
containing 0 such that 
is a diffeomorphism.
That is, there is a smooth inverse 
.
Inverse Function Theorem
See also
Diffeomorphism, Implicit Function Theorem, JacobianThis entry contributed by Todd Rowland
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Cite this as:
Weisstein, Eric W., with contributions by Todd Rowland. "Inverse Function Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/InverseFunctionTheorem.html