Brouwer fixed-point theorem: Difference between revisions

Revision as of 21:08, 14 March 2022

Revision as of 21:09, 14 March 2022

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The theorem holds only for functions that are endomorphisms (or have the same set as the ___domain and range) and for sets that are ''compact'' (thus, in particular, bounded and closed) and ''convex'' (or [[Homeomorphism|homeomorphic]] to convex). The following examples show why the pre-conditions are important.

'''===The Function <math>f</math> as an Endomorphism==='''

Consider the function

:<math>f(x) = x+1</math>

with ___domain [-1,1]. The range of the function is [-2,0]. Thus f is not an endomorphism.

===Boundedness===