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

[cateater2000]

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

 

[HallsofIvy]

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?