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, then it is an invertible function near p. That is, an inverse function to f 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.

The inverse function theorem can be generalized to differentiable maps between differentiable manifolds. In this context the theorem states that for a differentiable map f : M → N, if the derivative of f, (Df)_p : T_pM → T_f(p)N is a linear isomorphism at a point p in M then there exists an open neighborhood U of p such that f|_U : U → f(U) is a diffeomorphism. Note that this implies that M and N must have the same dimension.

If the derivative of f is an isomorphism at all points p in M then the map f is a local diffeomorphism.