Theorem

Inverse Function Theorem

Analysis

A continuously differentiable function whose derivative (or Jacobian) is invertible at a point is itself invertible on some neighborhood of that point, with a differentiable inverse. It is closely related to, and often proved alongside, the implicit function theorem.

Facts

Statement

The inverse function theorem gives sufficient conditions under which a function has a local inverse near a point, namely that the function's best linear approximation, its derivative, is itself invertible there. 1

Classification

Statement Form

Existence Theorem 1

Connections

Source Inverse Function Theorem (Wikipedia)

Émile Picard

Source Inverse Function Theorem (Wikipedia)

Sources

1. Inverse Function Theorem (Wikipedia)

Wikimedia Foundation

Lead section and Statements section
In mathematical analysis, the inverse function theorem gives sufficient conditions for a function to have an inverse function.
Proved By: Édouard Goursat, Lead section
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.
Proved By: Émile Picard, Lead section
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.

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Inverse Function Theorem

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