Can Non-Continuous Functions Have Fixed Points on Compact Convex Sets?

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The discussion centers on the assertion that non-continuous functions cannot have fixed points when mapping from a nonempty compact convex set K in R^n to itself. A participant challenges this claim by providing an example of a non-continuous function defined on R^1, specifically f(1)=2, f(2)=1, and f(x)=x for all other x. This example demonstrates that non-continuous functions can indeed map compact convex sets and raises questions about the validity of the original statement.

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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
 
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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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