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===Selections===
When <math>f: \mathbb{R}^n \to \mathbb{R}^m</math> with <math>m\leq n</math>, <math>f</math> is <math>k</math> times [[continuously differentiable]], and the Jacobian <math>A=\nabla f(\overline{x})</math> at a point <math>\overline{x}</math> is of [[rank (linear algebra)|rank]] <math>m</math>, the inverse of <math>f</math> may not be unique. However, there exists a local [[Choice function#Choice function of a multivalued map|selection function]] <math>s</math> such that <math>f(s(y)) = y</math> for all <math>y</math> in a [[neighborhood (mathematics)|neighborhood]] of <math>\overline{y} = f(\overline{x})</math>, <math>s(\overline{y}) = \overline{x}</math>, <math>s</math> is <math>k</math> times continuously differentiable in this neighborhood, and <math>\nabla s(\overline{y}) = A^T(A A^T)^{-1}</math> (<math>\nabla s(\overline{y})</math> is the [[Moore–Penrose pseudoinverse]] of <math>A</math>).<ref>{{cite book |last1=Dontchev |first1=Asen L. |last2=Rockafellar |first2=R. Tyrrell |title=Implicit Functions and Solution Mappings: A View from Variational Analysis |date=2014 |publisher=Springer-Verlag |___location=New York |isbn=978-1-4939-1036-6 |page=54 |edition=Second}}</ref>
=== Over a real closed field ===
The inverse function theorem also holds over a [[real closed field]] (or more generally an [[O-minimal structure]]).<ref>Theorem 2.11. in {{cite book |doi=10.1017/CBO9780511525919|title=Tame Topology and O-minimal Structures. London Mathematical Society lecture note series, no. 248|year=1998 |last1=Dries |first1=L. P. D. van den |authorlink = Lou van den Dries|isbn=9780521598385|publisher=Cambridge University Press|___location=Cambridge, New York, and Oakleigh, Victoria }}</ref>
==See also==
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