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===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)
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.
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