Inverse function theorem - Definition

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 states that 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. That is, an inverse function exists, in some neighborhood of F(P).

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