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If {{mvar|'''w'''}} is only a ''continuous'' unit tangent vector on {{mvar|''S''}}, by the [[Weierstrass approximation theorem]], it can be uniformly approximated by a polynomial map {{mvar|'''u'''}} of {{mvar|''A''}} into Euclidean space. The orthogonal projection on to the tangent space is given by {{mvar|'''v'''}}({{mvar|'''x'''}}) = {{mvar|'''u'''}}({{mvar|'''x'''}}) - {{mvar|'''u'''}}({{mvar|'''x'''}}) ⋅ {{mvar|'''x'''}}. Thus {{mvar|'''v'''}} is polynomial and nowhere vanishing on {{mvar|''A''}}; by construction {{mvar|'''v'''}}/||{{mvar|'''v'''}}|| is a smooth unit tangent vector field on {{mvar|''S''}}, a contradiction.
The continuous version of the hairy ball theorem can now be used to prove the Brouwer fixed point theorem. First suppose that {{mvar|''n''}} is
:<math>{\mathbf w}({\mathbf x}) = (1 - {\mathbf x}\cdot {\mathbf f}({\mathbf x}))\, {\mathbf x} - (1 - {\mathbf x}\cdot {\mathbf x})\, {\mathbf f}({\mathbf x}).</math>
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:<math>{\mathbf X}({\mathbf x},t)=(-t\,{\mathbf w}({\mathbf x}), {\mathbf x}\cdot {\mathbf w}({\mathbf x})).</math>
By construction {{mvar|'''X'''}} is a continuous vector field on the unit sphere of {{mvar|''W''}}, satisfying the tangency condition {{mvar|'''y'''}} ⋅ {{mvar|'''X'''}}({{mvar|'''y'''}}) = 0. Moreover, {{mvar|'''X'''}}({{mvar|'''y'''}}) is nowhere vanishing (because, if {{var|'''x'''}} has norm 1, then {{mvar|'''x'''}} ⋅ {{mvar|'''w'''}}({{mvar|''x''}}) is non-zero; while if {{mvar|'''x'''}} has norm strictly less than 1, then {{mvar|''t''}} and {{mvar|'''w'''}}({{mvar|'''x'''}}) are both non-zero). This contradiction proves the fixed point theorem when {{mvar|''n''}} is
The advantage of this proof is that it uses only elementary techniques; more general results like the [[Borsuk-Ulam theorem]] require tools from [[algebraic topology]].<ref name="Milnor78">{{harvnb|Milnor|1978}}</ref>
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