Editing Brouwer fixed-point theorem (section)

===A proof using degree===
Brouwer's original 1911 proof relied on the notion of the [[degree of a continuous mapping]], stemming from ideas in [[differential topology]]. Several modern accounts of the proof can be found in the literature, notably {{harvtxt|Milnor|1965}}.<ref name="Milnor">{{harvnb|Milnor|1965|pages=1–19}}</ref><ref>{{cite book | last = Teschl| first = Gerald| author-link = Gerald Teschl| title = Topics in Linear and Nonlinear Functional Analysis|url=https://www.mat.univie.ac.at/~gerald/ftp/book-fa/fa.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.mat.univie.ac.at/~gerald/ftp/book-fa/fa.pdf |archive-date=2022-10-09 |url-status=live|chapter=10. The Brouwer mapping degree|access-date=1 February 2022|year=2019|publisher=[[American Mathematical Society]]|series=Graduate Studies in Mathematics}}</ref>

Let <math>K=\overline{B(0)}</math> denote the closed unit ball in <math>\mathbb R^n</math> centered at the origin.  Suppose for simplicity that <math>f:K\to K</math> is continuously differentiable.  A [[regular value]] of <math>f</math> is a point <math>p\in B(0)</math> such that the [[Jacobian matrix and determinant|Jacobian]] of <math>f</math> is non-singular at every point of the preimage of <math>p</math>.  In particular, by the [[inverse function theorem]], every point of the preimage of <math>f</math> lies in <math>B(0)</math> (the interior of <math>K</math>).  The degree of <math>f</math> at a regular value <math>p\in B(0)</math> is defined as the sum of the signs of the [[Jacobian determinant]] of <math>f</math> over the preimages of <math>p</math> under <math>f</math>:

:<math>\operatorname{deg}_p(f) = \sum_{x\in f^{-1}(p)} \operatorname{sign}\,\det (df_x).</math>

The degree is, roughly speaking, the number of "sheets" of the preimage ''f'' lying over a small open set around ''p'', with sheets counted oppositely if they are oppositely oriented.  This is thus a generalization of [[winding number]] to higher dimensions.

The degree satisfies the property of ''homotopy invariance'': let <math>f</math> and <math>g</math> be two continuously differentiable functions, and <math>H_t(x)=tf+(1-t)g</math> for <math>0\le t\le 1</math>.  Suppose that the point <math>p</math> is a regular value of <math>H_t</math> for all ''t''.  Then <math>\deg_p f = \deg_p g</math>.

If there is no fixed point of the boundary of <math>K</math>, then the function 
:<math>g(x)=\frac{x-f(x)}{\sup_{y\in K}\left|y-f(y)\right|}</math>

is well-defined, and

<math>H(t,x) = \frac{x-tf(x)}{\sup_{y\in K}\left|y-tf(y)\right|}</math>

defines a homotopy from the identity function to it.  The identity function has degree one at every point.  In particular, the identity function has degree one at the origin, so <math>g</math> also has degree one at the origin.  As a consequence, the preimage <math>g^{-1}(0)</math> is not empty.  The elements of <math>g^{-1}(0)</math> are precisely the fixed points of the original function ''f''.

This requires some work to make fully general.  The definition of degree must be extended to singular values of ''f'', and then to continuous functions.  The more modern advent of [[homology theory]] simplifies the construction of the degree, and so has become a standard proof in the literature.