Brouwer fixed-point theorem: Difference between revisions

Revision as of 21:52, 1 January 2023

Revision as of 03:02, 23 February 2023

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Consider the function

:<math>f(x) = \frac{x+1}{2},</math>

which is a continuous function from the open interval (−1,1) to itself. InSince thisx interval,= it1 shiftsis everynot pointpart toof the rightinterval, sothere itis cannot havenot a fixed point of f(x) = x. The space (−1,1) is convex and bounded, but not closed. TheOn the other hand, the function ''f'' {{em|does}} have a fixed point for the closed interval [−1,1], namely ''f''(1) = 1.

===Convexity===