Inverse function theorem

Counter-example

Lemma—Let g:B(0,r)→ℝn a map with a constant x,y in B(0,r), we have B(0,(1−c)r)⊂f(B(0,r))⊂B(0,(1+c)r.)

More generally, the statement remains true if f′(0)=I. Let |g(y)−g(x)|≤|y−x|sup0<t<1|g′(x+t(y−x))|. Since r>0 such that B(0,r). Then the early lemma says that B(0,r/2)⊂f(B(0,r)). Then g=f−1 denote the inverse of x=g(y), we write |h−k|=|f(x+h)−f(x)−h|≤|h|/2 and so |g(y+k)−g(y)−A−1k|=|h−A−1(f(x+h)−f(x))|≤‖A−1‖|Ah−f(x+h)+f(x)|. As |h|/|k| is bounded. Hence, g′(y)=f′(g(y))−1. Also, ι:T↦T−1; so (1−c)|x−y|≤|f(x)−f(y)|. This proves the first part. Next, we show B(0,(1−c)r), find a fixed point of the map |y|≤(1−c)r′ and the bar means a closed ball. To find a fixed point, we use the contraction mapping theorem and checking that |f(x)|=|x+g(x)−g(0)|≤(1+c)|x|.◻

As might be clear, this proof is not substantially different from the previous one, as the proof of the contraction mapping theorem is by successive approximation.

The inverse function theorem can be used to solve a system of equations x=(x1,…,xn), provided the Jacobian matrix is invertible. The implicit function theorem allows to solve a more general system of equations: x. Though more general, the theorem is actually a consequence of the inverse function theorem. First, the precise statement of the implicit function theorem is as follows:

• given a map f is continuously differentiable in a neighborhood of b is invertible, then there exists a differentiable map a,b such that y=g(x); i.e., F:U×V→W has the inverse (x,y)=F(G1(x,y),G2(x,y))=(G1(x,y),f(G1(x,y),G2(x,y))), implying g(x)=G2(x,0) has the required property. ℝn (since the result is local, there is no loss of generality with considering such a map). Fix a point ℝn, assume the matrix F:U→ℝr×ℝn−r=ℝn,x↦(f(x),xr+1,…,xn) is such that G of (b,ar+1,…,an). We then have G, g=G(b,⋅):W→f−1(b)∩U′,(xr+1,…,xn)↦G(b,xr+1,…,xn) is bijective with the smooth inverse. That is to say, a. Hence, f:P→E is transversal to a submanifold f is locally bijective (or locally diffeomorphic of some class). The next topological lemma can be used to upgrade local injectivity to injectivity that is global to some extent.
  One of Spivak's books (Editorial note: give the exact location).</ref>{{full citation needed|date=August 2023}}<ref>{{harvnb|Hirsch|1976|loc=Ch. 2, § 1., Exercise 7.}} NB: This one is for a <math>C^1</math>-immersion.</ref> If <math>A</math> is a closed subset of a (second-countable) topological manifold <math>X</math> (or, more generally, a topological space admitting an [[exhaustion by compact subsets]]) and <math>f : X \\to Z</math>, <math>Z</math> some topological space, is a local homeomorphism that is injective on <math>A</math>, then <math>f</math> is injective on some neighborhood of <math>A</math>."}},"i":0}}]}'>X (or, more generally, a topological space admitting an exhaustion by compact subsets) and A, then X is compact. If the conclusion of the theorem is false, we can find two sequences xi,yi each converge to some points f is injective on i is large enough, f is injective; thus, S×S for any subset X1⊂X2⊂⋯ be an increasing sequence of compact subsets with union Xi+1. Then, by the first part of the proof, for each A∩Xi such that ◻ (See also for an alternative approach.)

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

  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 on 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>)."}},"i":0}}]}'>ℝn or more generally of manifolds. Assume f is injective on a closed subset A, then A and A is a point, then the above is the usual inverse function theorem.

  Holomorphic inverse function theorem

  edit

  There is a version of the inverse function theorem for holomorphic maps.

  {{harvnb|Griffiths|Harris|1978|loc=p. 18.}}</ref><ref>{{cite book |first1=K. |last1=Fritzsche |first2=H. |last2=Grauert |author-link2=Hans Grauert |title=From Holomorphic Functions to Complex Manifolds |publisher=Springer |year=2002 |pages=33–36 |isbn=978-0-387-95395-3 |url=https://books.google.com/books?id=jSeRz36zXIMC&pg=PA33 }}</ref> Let <math>U, V \\subset \\mathbb{C}^n</math> be open subsets such that <math>0 \\in U</math> and <math>f : U \\to V</math> a holomorphic map whose Jacobian matrix in variables <math>z_i, \\overline{z}_i</math> is invertible (the determinant is nonzero) at <math>0</math>. Then <math>f</math> is injective in some neighborhood <math>W</math> of <math>0</math> and the inverse <math>f^{-1} : f(W) \\to W</math> is holomorphic."}},"i":0}}]}'>0∈U and 0. Then 0 and the inverse f in variables zj,z‾j. Then we have 0 with continuously differentiable inverse. By chain rule, with fj−1∘f and k. ℂn, then C1), if the differential of p in p such that N containing p and F(p) have the same dimension, as is already directly implied from the assumption that dFp is an isomorphism. If the derivative of M then the map Y.

  Let F:U→Y a continuously differentiable function, and assume that the Fréchet derivative X onto F(0) in F(G(y))=y for all G(y) is the only sufficiently small solution X⊂C1([0,1]) of continuously-differentiable real-valued functions on the unit interval such that F:X→Y,F(u)=u′+u2. Then DF(u)h=h′+2uh. At h↦h′⁠. This is an isomorphism from ⁠⁠0⁠. In particular, for every sufficiently small u′(t)+u(t)2=g(t),u(0)=0, has a unique small solution ⁠p∈M, then there are open neighborhoods V of v:TF(p)N→V such that v−1∘F∘u. That is, p∈M such that the rank is constant in a neighborhood of F is injective (resp. surjective) at a point F is constant on that neighborhood, and the constant rank theorem applies.

  Polynomial functions

  edit
  f:ℝn→ℝm with k times continuously differentiable, and the Jacobian m, the inverse of f(s(y))=y for all s(y‾)=x‾, ∇s(y‾)=AT(AAT)−1 (kn that is continuously differentiable.

  The usual proof of the IFT uses Banach's fixed point theorem, which relies on the Cauchy completeness. That part of the argument is replaced by the use of the extreme value theorem, which does not need completeness. Explicitly, in § A proof using the contraction mapping principle, the Cauchy completeness is used only to establish the inclusion y in B‾(0,r). If f(x)=y, since x0 on the closed ball 2−1|x|≤|f(x)|. Since ◻

  Alternatively, one can deduce the theorem from the one over real numbers by Tarski's principle.Lebl, Jiří. "4.4 Inverse function theorem". Basic Analysis I: Introduction to Real Analysis. Retrieved 17 June 2026.

• 1 2 3 Theorem 1.1.7. in Tao, Terence (12 September 2011). "The inverse function theorem for everywhere differentiable maps". Retrieved 26 July 2019.
• ↑ Cartan, Henri (1971). Calcul Differentiel (in French). Hermann. pp. 55–61. ISBN 978-0-395-12033-0.
• ↑ Theorem 17.7.2 in Luenberger, David G. (1969). Optimization by Vector Space Methods. New York: John Wiley & Sons. pp. 240–242. ISBN 0-471-55359-X.
• ↑ Boothby, William M. (1986). An Introduction to Differentiable Manifolds and Riemannian Geometry (Second ed.). Orlando: Academic Press. pp. 46–50. ISBN 0-12-116052-1.
• ↑ Allendoerfer, Carl B. (1974). "Theorems about Differentiable Functions". Calculus of Several Variables and Differentiable Manifolds. New York: Macmillan. pp. 54–88. ISBN 0-02-301840-2.
• Griffiths, Phillip; Harris, Joseph (1978), Principles of Algebraic Geometry, John Wiley & Sons, ISBN 978-0-471-05059-9.
• Renardy, Michael; Rogers, Robert C. (2004). An Introduction to Partial Differential Equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. pp. 337–338. ISBN 0-387-00444-0.