Notes on the Implicit Function Theorem

  • KC Border
  • Published 2013

Abstract

The use of zero in the above equation serves to simplify notation. The condition f(x, p) = y is equivalent to g(x, p) = 0 where g(x, p) = f(x, p) − y, and this transformation of the problem is common in practice. The implicit function theorem gives conditions under which it is possible to solve for x as a function of p in the neighborhood of a known solution (x̄, p̄). There are actually many implicit function theorems. If you make stronger assumptions, you can derive stronger conclusions. In each of the theorems that follows we are given a subset X of Rn, a metric space P (of parameters), a function f from X×P into Rn, and a point (x̄, p̄) in the interior of X×P such that Dxf(x̄, p̄) exists and is invertible. Each asserts the existence of neighborhoods U of x̄ and W of p̄ and a function ξ : W → U such that f ( ξ(p), p ) = f(x̄, p̄) for all p ∈ W . They differ in whether ξ is uniquely defined (in U) and how smooth it is. The following table serves as a guide to the theorems. For ease of reference, each theorem is stated as a standalone result.

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Cite this paper

@inproceedings{Border2013NotesOT, title={Notes on the Implicit Function Theorem}, author={KC Border}, year={2013} }