A Lipschitz Implicit Function Theorem.

Question

I look for a reference (book or article) that contains the statement of a version of the implicit function theorem as stated below. This statement I found in notes (with due proof) on the implicit function written by KC Border.

In these notes the author states that the proof is based on the proof of the implicit function theorem for two-variable functions given by Apostol in his calculus book. I looked at the proof given by Apostol in his calculus book (Calculus vol.2) and in fact the proof method of the above theorem is, with due adaptations, the same.

I am writing notes on the implicit function theorem. However, I did not want to have to repeat the proof, but to indicate to the reader a formal reference containing the proof.

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Update 1 13 November, 2018

I have already searched in the following textbooks

• Implicit Functions and Solution Mappings. A View from Variational Analysis

• The Implicit Function Theorem: History, Theory, and Applications

• Nonlinear Functional Analysis

But I did not succeed.

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Update 2 January 27, 2021

I originally found this theorem in the Book Aplicações da Topologia à Análise (Applications Of Topology To Analysis) in Portuguese. But I'd like it to be a reference in English language.

I have already searched in Treatise on Analysis. Foundations of modern analysis. vol. I by J. Dieudonne but again I didn't find anything like that.