Editing Brouwer fixed-point theorem (section)

===A proof by Hirsch===
There is also a quick proof, by [[Morris Hirsch]], based on the impossibility of a differentiable retraction.  Let ''f'' denote a continuous map from the unit ball D<sup>n</sup> in n-dimensional Euclidean space to itself and assume that ''f'' fixes no point. By continuity and the fact that D<sup>n</sup> is compact, it follows that for some ε > 0, ∥x - ''f''(x)∥ > ε for all x in D<sup>n</sup>. Then the map ''f'' can be approximated by a smooth map retaining the property of not fixing a point; this can be done by using the [[Weierstrass approximation theorem]] or by [[convolution|convolving]] with smooth [[bump function]]s.  One then defines a retraction as above by sending each x to the point of ∂D<sup>n</sup> where the unique ray from x through ''f''(x) intersects ∂D<sup>n</sup>, and this must now be a differentiable mapping.  Such a retraction must have a non-singular value p ∈ ∂D<sup>n</sup>, by [[Sard's theorem]], which is also non-singular for the restriction to the boundary (which is just the identity).  Thus the inverse image ''f''<sup> -1</sup>(p) would be a compact 1-manifold with boundary. Such a boundary would have to contain at least two endpoints, and these would have to lie on the boundary of the original ball. This would mean that the inverse image of one point on ∂D<sup>n</sup> contains a different point on ∂D<sup>n</sup>, contradicting the definition of a retraction D<sup>n</sup> → ∂D<sup>n</sup>.<ref>{{harvnb|Hirsch|1988}}</ref>

R. Bruce Kellogg, Tien-Yien Li, and [[James A. Yorke]] turned Hirsch's proof into a [[Computability|computable]] proof by observing that the retract is in fact defined everywhere except at the fixed points.{{sfn|Kellogg|Li|Yorke|1976}}  For almost any point ''q'' on the boundary — assuming it is not a fixed point — the 1-manifold with boundary mentioned above does exist and the only possibility is that it leads from ''q'' to a fixed point. It is an easy numerical task to follow such a path from ''q'' to the fixed point so the method is essentially computable.{{sfn|Chow|Mallet-Paret|Yorke|1978}} gave a conceptually similar path-following version of the homotopy proof which extends to a wide variety of related problems.