Inverse function theorem: Difference between revisions

Revision as of 09:58, 7 November 2024

Revision as of 10:01, 7 November 2024

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The lemma implies the following (a sort of) global version of the inverse function theorem:

{{math_theorem|name=Inverse function theorem|math_statement=<ref>Ch. I., § 3, Exercise 10. and § 8, Exercise 14. in V. Guillemin, A. Pollack. "Differential Topology". Prentice-Hall Inc., 1974. ISBN 0-13-212605-2.</ref> Let <math>f : U \to V</math> be a map between open subsets of <math>\mathbb{R}^n</math> or more generally of manifolds. Assume <math>f</math> is continuously differentiable (or is <math>C^k</math>). If <math>f</math> is injective on a closed subset <math>A \subset U</math> and if the Jacobian matrix of <math>f</math> is invertible at each point of <math>A</math>, then <math>f</math> is injective in a neighborhood <math>A'</math> of <math>A</math> and <math>f^{-1} : f(A') \to A'</math> is continuously differentiable (or is <math>C^k</math>).}}

Note that if <math>A</math> is a point, then the above is the usual inverse function theorem.