Inverse function theorem

In mathematical analysis, the inverse function theorem gives sufficient conditions for a function to have an inverse function. The essential idea is that if the best linear approximation to the function at a point is invertible, then with sufficient regularity assumptions, the function should also be invertible near that point. In its simplest form, the theorem states that if a real function f is differentiable in an open interval, with a continuous derivative, then in a neighborhood of any point where the derivative is not zero, f has an inverse function. The inverse function is also continuously differentiable, and the inverse function rule expresses its derivative as the multiplicative inverse of the derivative of f. The theorem applies verbatim to complex-valued functions of a complex variable. It generalizes to functions from n-tuples (of real or complex numbers) to n-tuples, and to functions between vector spaces of the same finite dimension, by replacing "derivative" with "Jacobian matrix" and "nonzero derivative" with "nonzero Jacobian determinant". If the function of the theorem belongs to a higher differentiability class, the same is true for the inverse function. There are also versions of the inverse function theorem for holomorphic functions, for differentiable maps between manifolds, for differentiable functions between Banach spaces, and so forth. The theorem was first established by Picard and Goursat using an iterative scheme: the basic idea is to prove a fixed point theorem using the contraction mapping theorem.