# local inverse of analytic function

Let $$f:R^n\to R^n$$ be an analytic map such that $$df_a$$ is invertible. The "usual" local inversion theorem says that $$f$$ is locally invertible and that the inverse will be smooth.

I'm quite sure that, if $$f$$ is analytic, the inverse will be analytic too, but I cannot prove it.

I know a proof in the case $$n=1$$ trough complex analysis.

This is almost a duplicate of this question with two differences :

• I don't mind passing trough complex analysis
• I need a statement on an open set of $$R^n$$, not an interval of R.
• I want a proof or a link, not a reference to a book (I do not have access to a math library)
• The proof through complex analysis can be carried through also in more variables: extend $f$ to a neighborhood of $\mathbf{R}^n\subset\mathbf{C}^n$, prove that the differential of the extension is invertible at every point, and prove that the inverse of a holomorphic function (in several variables) is holomorphic Commented Dec 30, 2021 at 19:36
• See exercise 14 and 15 on page 226 here: mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/…. Commented Dec 30, 2021 at 21:06
• Also this one : books.google.com/books?id=jSeRz36zXIMC&pg=PA33 (got from wikipedia) Commented Dec 30, 2021 at 21:12