Editing Implicit function theorem (section)

== Generalizations ==

=== Banach space version ===
Based on the [[inverse function theorem]] in [[Banach space]]s, it is possible to extend the implicit function theorem to Banach space valued mappings.<ref>{{Cite book |last=Lang |first=Serge |author-link=Serge Lang |title=Fundamentals of Differential Geometry |url=https://archive.org/details/fundamentalsdiff00lang_678 |url-access=limited |year=1999 |publisher=Springer | location=New York |series=Graduate Texts in Mathematics |isbn=0-387-98593-X |pages=[https://archive.org/details/fundamentalsdiff00lang_678/page/n15 15]–21 }}</ref><ref>{{Cite book |last=Edwards |first=Charles Henry |title=Advanced Calculus of Several Variables |publisher=Dover Publications |location=Mineola, New York |year=1994 |orig-year=1973 |isbn=0-486-68336-2 |pages=417–418 }}</ref>

Let ''X'', ''Y'', ''Z'' be [[Banach space]]s. Let the mapping {{math|''f'' : ''X'' × ''Y'' → ''Z''}} be continuously [[Fréchet differentiable]]. If <math>(x_0,y_0)\in X\times Y</math>, <math>f(x_0,y_0)=0</math>, and <math>y\mapsto Df(x_0,y_0)(0,y)</math> is a Banach space isomorphism from ''Y'' onto ''Z'', then there exist neighbourhoods ''U'' of ''x''<sub>0</sub> and ''V'' of ''y''<sub>0</sub> and a Fréchet differentiable function ''g'' : ''U'' → ''V'' such that ''f''(''x'', ''g''(''x'')) = 0 and ''f''(''x'', ''y'') = 0 if and only if ''y'' = ''g''(''x''), for all <math>(x,y)\in U\times V</math>.

=== Implicit functions from non-differentiable functions ===
Various forms of the implicit function theorem exist for the case when the function ''f'' is not differentiable. It is standard that local strict monotonicity suffices in one dimension.<ref>{{springer |title=Implicit function |id=i/i050310 |last=Kudryavtsev |first=Lev Dmitrievich }}</ref> The following more general form was proven by Kumagai based on an observation by Jittorntrum.<ref>{{Cite journal |first=K. |last=Jittorntrum |title=An Implicit Function Theorem |journal=Journal of Optimization Theory and Applications |volume=25 |issue=4 |year=1978 |doi=10.1007/BF00933522 |pages=575–577 |s2cid=121647783 }}</ref><ref>{{Cite journal |first=S. |last=Kumagai |title=An implicit function theorem: Comment |journal=Journal of Optimization Theory and Applications |volume=31 |issue=2 |year=1980 |doi=10.1007/BF00934117 |pages=285–288 |s2cid=119867925 }}</ref>

Consider a continuous function <math>f : \R^n \times \R^m \to \R^n</math> such that <math>f(x_0, y_0) = 0</math>. If there exist open neighbourhoods <math>A \subset \R^n</math> and <math>B \subset \R^m</math> of ''x''<sub>0</sub> and ''y''<sub>0</sub>, respectively, such that, for all ''y'' in ''B'', <math>f(\cdot, y) : A \to \R^n</math> is locally one-to-one, then there exist open neighbourhoods <math>A_0 \subset \R^n</math> and <math>B_0 \subset \R^m</math> of ''x''<sub>0</sub> and ''y''<sub>0</sub>, such that, for all <math>y \in B_0</math>, the equation
''f''(''x'', ''y'') = 0 has a unique solution
<math display="block">x = g(y) \in A_0,</math>
where ''g'' is a continuous function from ''B''<sub>0</sub> into ''A''<sub>0</sub>.

=== Collapsing manifolds ===
Perelman’s collapsing theorem for [[3-manifold]]s, the capstone of his proof of Thurston's [[geometrization conjecture]], can be understood as an extension of the implicit function theorem.<ref>{{cite journal |last1=Cao |first1=Jianguo |last2=Ge |first2=Jian |title=A simple proof of Perelman's collapsing theorem for 3-manifolds |journal=J. Geom. Anal. |date=2011 |volume=21 |issue=4 |pages=807–869|doi=10.1007/s12220-010-9169-5 |arxiv=1003.2215 |s2cid=514106 }}</ref>