Implicit Function Theorem

A local statement guaranteeing that, near a point where a system of differentiable equations F(x,y)=0 has a nondegenerate Jacobian with respect to y, there exists a differentiable map y = g(x) solving the system in a neighborhood of that point.

A local result that guarantees, under differentiability and a nonsingular Jacobian condition with respect to certain variables, the ability to solve an equation F(x,y)=0 for y as a differentiable function y=g(x) near a base point.