Inverse function theorem: Difference between revisions

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{{Calculus}}

In [[real analysis]], a branch of [[mathematics]], the '''inverse function theorem''' is a [[theorem]] that asserts that, if a [[real function]] ''f'' has a [[continuously differentiable function|continuous derivative]] near a point where its derivative is nonzero, then, near this point, ''f'' has an [[inverse function]]. The inverse function is also [[differentiable function|differentiable]], and the ''[[inverse function rule]]'' expresses its derivative as the [[multiplicative inverse]] of the derivative of ''f''.

The theorem applies verbatim to [[complex-valued function]]s of a [[complex number|complex variable]]. It generalizes to functions from

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=== Over a real closed field ===

The inverse function theorem also holds over a [[real closed field]] ''k'' (or an [[Oo-minimal structure]]).<ref>Chapter 7, Theorem 2.11. in {{cite book |doi=10.1017/CBO9780511525919|title=Tame Topology and O-minimal Structures. London Mathematical Society lecture note series, no. 248|year=1998 |last1=Dries |first1=L. P. D. van den |authorlink = Lou van den Dries|isbn=9780521598385|publisher=Cambridge University Press|___location=Cambridge, New York, and Oakleigh, Victoria }}</ref> Precisely, the theorem holds for a semialgebraic (or definable) map between open subsets of <math>k^n</math> 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 {{section link||A_proof_using_the_contraction_mapping_principle}}, the Cauchy completeness is used only to establish the inclusion <math>B(0, r/2) \subset f(B(0, r))</math>. Here, we shall directly show <math>B(0, r/4) \subset f(B(0, r))</math> instead (which is enough). Given a point <math>y</math> in <math>B(0, r/4)</math>, consider the function <math>P(x) = |f(x) - y|^2</math> defined on a neighborhood of <math>\overline{B}(0, r)</math>. If <math>P'(x) = 0</math>, then <math>0 = P'(x) = 2[f_1(x) - y_1 \cdots f_n(x) - y_n]f'(x)</math> and so <math>f(x) = y</math>, since <math>f'(x)</math> is invertible. Now, by the extreme value theorem, <math>P</math> admits a minimal at some point <math>x_0</math> on the closed ball <math>\overline{B}(0, r)</math>, which can be shown to lie in <math>B(0, r)</math> using <math>2^{-1}|x| \le |f(x)|</math>. Since <math>P'(x_0) = 0</math>, <math>f(x_0) = y</math>, which proves the claimed inclusion. <math>\square</math>