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Suppose, for contradiction, that a continuous function {{nowrap|''f'' : ''D''<sup>''n''</sup> → ''D''<sup>''n''</sup>}} has ''no'' fixed point. This means that, for every point x in ''D''<sup>''n''</sup>, the points ''x'' and ''f''(''x'') are distinct. Because they are distinct, for every point x in ''D''<sup>''n''</sup>, we can construct a unique ray from ''f''(''x'') to ''x'' and follow the ray until it intersects the boundary ''S''<sup>''n''−1</sup> (see illustration). By calling this intersection point ''F''(''x''), we define a function ''F'' : ''D''<sup>''n''</sup> → ''S''<sup>''n''−1</sup> sending each point in the disk to its corresponding intersection point on the boundary. As a special case, whenever x itself is on the boundary, then the intersection point ''F''(''x'') must be ''x''.
Consequently, ''F'' is a special type of continuous function known as a [[retraction (topology)|retraction]]: every point of the [[codomain]] (in this case ''S''<sup>''n''−1</sup>) is a fixed point of ''F''.
Intuitively it seems unlikely that there could be a retraction of ''D''<sup>''n''</sup> onto ''S''<sup>''n''−1</sup>, and in the case ''n'' = 1, the impossibility is more basic, because ''S''<sup>0</sup> (i.e., the endpoints of the closed interval ''D''<sup>1</sup>) is not even connected. The case ''n'' = 2 is less obvious, but can be proven by using basic arguments involving the [[fundamental group]]s of the respective spaces: the retraction would induce a surjective [[group homomorphism]] from the fundamental group of ''D''<sup>2</sup> to that of ''S''<sup>1</sup>, but the latter group is isomorphic to '''Z''' while the first group is trivial, so this is impossible. The case ''n'' = 2 can also be proven by contradiction based on a theorem about non-vanishing [[vector field]]s.
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