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Convexity and fixed points

  1. Mar 28, 2006 #1
    Suppose that K is a nonempty compact convex set in R^n. If f:K->K is not continuous, then f will not have any fixed point.


    I believe this statement is false, but I cannot think of a function(not continuous) that maps a compact convex set to another compact convex set.

    any tips would be appreciated
     
  2. jcsd
  3. Mar 29, 2006 #2

    HallsofIvy

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    There are a lot of non-continuous functions! For example, in R1, define f(1)= 2, f(2)= 1, f(x)= x for any x other than 1 or 2. That works doesn't it?
     
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