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 [[functiondifferentiable (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 function|continuous]] and non-zero at the point''. The theorem also gives a [[formula]] for the [[derivative]] of the resulting inverse function, called [[inverse function rule]].

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 [[holomorphic function]]s, for differentiable maps between [[manifold]]s, for differentiable functions between [[Banach space]]s, and so forth.