Convexity and fixed points

[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?

 

[tacman]

There are indeed functions that are not continuous but still have fixed points on a compact convex set. One example is the function f(x) = 2x on the interval [0,1]. This function is not continuous at x = 0, but it still has a fixed point at x = 0.5.

Another example is the function f(x) = -x on the interval [-1,1]. This function is not continuous at x = 0, but it has a fixed point at x = 0.

In general, if a function is not continuous, it does not necessarily mean that it cannot have fixed points. It is possible for a function to have fixed points even if it is not continuous. However, the statement is true in the context of convexity. If a function is not continuous, it cannot have any fixed points on a nonempty compact convex set K. This is because a convex set is closed and bounded, and a non-continuous function will not preserve these properties, thus not being able to map a point in K to itself.

In summary, while there are functions that are not continuous but still have fixed points, in the context of convexity and compactness, a non-continuous function will not have any fixed points on a compact convex set.