Inverse function theorem: Difference between revisions

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It can happen that a function <math>f</math> may be injective near a point <math>a</math> while <math>f'(a) = 0</math>. An example is <math>f(x) = (x - a)^3</math>. In fact, for such a function, the inverse cannot be differentiable at <math>b = f(a)</math>, since if <math>f^{-1}</math> were differentiable at <math>b</math>, then, by the chain rule, <math>1 = (f^{-1} \circ f)'(a) = (f^{-1})'(b)f'(a)</math>, which implies <math>f'(a) \ne 0</math>. (The situation is different for holomorphic functions; see [[#Holomorphic inverse function theorem]] below.)

For functions of more than one variable, the theorem states that if {{Mvar|<math>f}}</math> is a continuously differentiable function from an open subset <math>A</math> of <math>\mathbb{R}^n</math> into <math>\R^n</math>, and the [[total derivative|derivative]] <math>f'(a)</math> is invertible at a point {{Mvar|a}} (that is, the determinant of the [[Jacobian matrix and determinant|Jacobian matrix]] of {{Mvar|f}} at {{Mvar|a}} is non-zero), then there exist neighborhoods <math>U</math> of <math>a</math> in <math>A</math> and <math>V</math> of <math>b = f(a)</math> such that <math>f(U) \subset V</math> and <math>f : U \to V</math> is bijective.<ref name="Hörmander">Theorem 1.1.7. in {{cite book|title=The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis|series=Classics in Mathematics|first=Lars|last= Hörmander|author-link=Lars Hörmander|publisher=Springer|year= 2015|edition=2nd|

isbn= 978-3-642-61497-2}}</ref> Writing <math>f=(f_1,\ldots,f_n)</math>, this means that the system of {{Mvar|n}} equations <math>y_i = f_i(x_1, \dots, x_n)</math> has a unique solution for <math>x_1, \dots, x_n</math> in terms of <math>y_1, \dots, y_n</math> when <math>x \in U, y \in V</math>. Note that the theorem ''does not'' say <math>f</math> is bijective onto the image where <math>f'</math> is invertible but that it is locally bijective where <math>f'</math> is invertible.