Editing Inverse function theorem (section)

{{Short description|Theorem in mathematics}}
{{Use dmy dates|date=December 2023}}
{{Calculus}}
In [[mathematics]], the '''inverse function theorem''' is a [[theorem]] that asserts that, if a [[real function]] ''f'' has a [[continuously differentiable function|continuous derivative]] near a point where its derivative is nonzero, then, near this point,  ''f'' has an [[inverse function]]. The inverse function is also [[differentiable function|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 function]]s of a [[complex number|complex variable]]. It generalizes to functions from 
''n''-[[tuples]] (of real or complex numbers) to ''n''-tuples, and to functions between [[vector space]]s 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 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]].