The implicit function theorem says: given a map f : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^m, if f(a, b) = 0, f is C^k in a neighborhood of (a, b) and the derivative of y \mapsto f(a, y) at b is invertible, then there exists a differentiable map g : U \to V for some neighborhoods U, V of a, b such that f(x, g(x)) = 0. The theorem follows from the inverse function theorem; see .
