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.  The [[indirect proof]] starts by noting that 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 which must now be differentiable.  Such a retraction must have a non-singular value, by [[Sard's theorem]], which is also non-singular for the restriction to the boundary (which is just the identity).  Thus the inverse image would be a 1-manifold with boundary. The boundary would have to contain at least two end points, both of which would have to lie on the boundary of the original ball—which is impossible in a retraction.<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 one 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.