Math 2.0

Real functions Corollary

Local homeomorphism from the inverse function theorem

Builds on Inverse function theoremContinuity in R^n

Statement

Let U⊆ℝnU \subseteq \mathbb{R}^n be open, and let f ⁣:U→ℝnf \colon U \to \mathbb{R}^n be of class C1C^1 with Df(x)Df(x) invertible for each x∈Ux \in U. Then each a∈Ua \in U has an open neighbourhood V⊆UV \subseteq U such that f(V)f(V) is open and ff restricted to VV is a homeomorphism onto f(V)f(V). In particular, ff is an open map: f(O)f(O) is open for each open set O⊆UO \subseteq U.

Remarks

This follows from the inverse function theorem, since a C1C^1 inverse is continuous. For each aa, the hypothesis is needed only at aa: if Df(a)Df(a) is invertible, ff is a homeomorphism from a neighbourhood of aa onto an open set. A local homeomorphism need not be injective. The map (x,y)↦(excos⁡y,exsin⁡y)(x, y) \mapsto (e^x \cos y, e^x \sin y) is a local homeomorphism from ℝ2\mathbb{R}^2 onto ℝ2∖{0}\mathbb{R}^2 \setminus \{0\}, but it is not injective.

Reading path

To understand this page from the ground up, read these 20 pages in order. The foundations come first.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFunction composition, concept, no evidence levelFunction compositionconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptContinuity in R^n, definition, no evidence levelContinuity in R^ndefinitionDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionChain rule in several variables, statement, no evidence levelChain rule in several…statementDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionJacobian matrix, definition, no evidence levelJacobian matrixdefinitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionDerivative of the inverse function, statement, no evidence levelDerivative of the inv…statementIterated derivative, definition, no evidence levelIterated derivativedefinitionJacobian determinant, definition, no evidence levelJacobian determinantdefinitionC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementInverse function theorem, statement, no evidence levelInverse function theo…statementLocal homeomorphism from the inverse function theorem, statement, no evidence levelLocal homeomorphism f…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Euclidean space object
  2. Function concept
  3. Function composition concept
  4. Fréchet derivative definition
  5. Function of several variables concept
  6. Limit of a function of several variables concept
  7. Partial derivative concept
  8. Continuity in R^n definition
  9. Differentiable function of several variables definition
  10. Partial derivative definition
  11. Chain rule in several variables statement
  12. Differentiable on an open set definition
  13. Jacobian matrix definition
  14. Higher partial derivatives definition
  15. Derivative of the inverse function statement
  16. Iterated derivative definition
  17. Jacobian determinant definition
  18. C^k function characterised by continuous derivatives statement
  19. Inverse function theorem statement
  20. Local homeomorphism from the inverse function theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Inverse_function_theorem, Several variables
About this page
Tag0Z8CSYVHDJ
URIm2://0Z8CSYVHDJ
Aliasesinvertible-derivative-implies-local-homeomorphism
Kindstatement, corollary
WritingW0 stub
MSC26B10
StatusThis copy of the site has no build data. Run m2 status 0Z8CSYVHDJ.
UsesInverse function theorem, Continuity in R^n

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