dissonant wrote:I often wonder about Brouwer's fixed point theorem. I.e "Every continuous function f from a closed disk to itself has at least one fixed point."

Say, by means of an analogy we are stirring a cup of coffee. At any two points in time there is some point which has not moved. Do these fixed points trace out some kind of "path"? I would think so. Intuitively, if the derivative is 0 at a point and the derivatives change continuously over the surface then if you are going to be a fixed point at the next point in time you would need to be arbitrarily close to the original.

Although, I would be happy if fixed point jumped around a lot too...

The points definitely don't move continuously. Consider rotating the coffee first about one point, than about any other point.