Inverse function theorem: Difference between revisions

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{{Calculus}}

In [[mathematics]], specifically [[differential calculus]], the '''inverse function theorem''' gives a [[Necessity and sufficiency|sufficient condition]] for a [[function (mathematics)|function]] to be [[Invertible function|invertible]] in a [[Neighbourhood (mathematics)|neighborhood]] of a point in its [[___domain of a function|___domain]]: namely, that its ''derivative is continuous and non-zero at the point''. The theorem also gives a [[formula]] for the [[derivative]] of the [[inverse function]].

In [[multivariable calculus]], this theorem can be generalized to any [[continuously differentiable]], [[vector-valued function]] whose [[Jacobian determinant]] is nonzero at a point in its ___domain, giving a formula for the [[Jacobian matrix]] of the inverse. There are also versions of the inverse function theorem for [[complex numbers|complex]] [[holomorphic function]]s, for differentiable maps between [[manifold]]s, for differentiable functions between [[Banach space]]s, and so forth.

The theorem was first established by [[Émile Picard|Picard]] and [[Édouard Goursat|Goursat]] using an iterative scheme: the basic idea is to prove a [[fixed point theorem]] using the [[contraction mapping theorem]].