Inverse function theorem

In mathematics, the inverse function theorem gives sufficient conditions for a vector valued function to be invertible on an open region containing a point in its domain.

The Theorem:

If at a point P a function f:Rⁿ-->Rⁿ has a Jacobian determinant that is nonzero, and F is continuously differentiable near P, it is an invertible function near P.

The Jacobian matrix of f^-1 at f(P) is then the inverse of Jf, evaluated at P.