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The Inverse Function Theorem. Let f : Rn −→ Rn be continuously differentiable on some open set containing a, and suppose det Jf(a) 6= 0. Then there is some open set V containing a and an open W containing f(a) such that f : V → W has a continuous inverse f−1 : W → V which is differentiable for all y ∈ W .
This chapter is devoted to the proof of the inverse and implicit function theorems. The inverse function theorem is proved in Section 1 by using the contraction mapping princi-ple. Next the implicit function theorem is deduced from the inverse function theorem in Section 2.
Lecture 12: The Inverse Function Theorem. Hart Smith. Department of Mathematics University of Washington, Seattle. Math 428, Winter 2020. Rouche’s Theorem. If f ; g are analytic on E, simple path in. ; g have no zeroes on , and. with int( ) E, (z) 1 for all z. f g, g(z) then: # zeroes of f in. = # zeroes of g in :
Theorem.If a function f has an inverse, then that inverse is unique. In particular, we have the right to talk about THE inverse of f, and the notation f −1 is not ambiguous.
Theorem 1. Suppose Ω ⊂ Rn is open, F : Ω → Rn is Ck, k ≥ 1, p0 ∈ Ω, q0 = F(p0). Suppose that DF(p0) is invertible. Then there is a neighborhood. U of p0 and a neighborhood V of q0 such that F : U → V is a bijection and F−1 : V → U is Ck. (One says that F is a Ck diffeomorphism.)
inverse function, gis an inverse function of f, so f is invertible. These theorems yield a streamlined method that can often be used for proving that a function is bijective and thus invertible.
Two examples illustrating the Inverse Function Theorem. 0.1. Example One. Let L(x) = x for x 2 R, a = 0,let p(x) = 8 >> >< >> >: 0 if x 1; x+1 if 1 < x 0; 1 x if 0 < x 1; 0 if 1 < x. Note that Lip(p) = 1 on any interval containing 0 and that = inffjL(x)j: jxj = 1g = 1: Let f(x) = L(x)+p(x). Since f is constant on [0;1] we see that f is not ...
