Pointless point?

I read this discussion on point on one of my firend's blog. I subscribe to his views but see it in a different perspective.

Any and all points can be fixed point or a floating point.

Consider Case 1:

Our petite point is a fixed point.

The fixed point mathematics say that a point that makes f(x)=x is a fixed point. If this fixed point has dimensions and is pregnent with other points then this argument fails, as then the function will map to many points inside a single point and become a one to many mapping. Since the function f(x) is one-to-one function this cannot be true. Ideally it should not be a mapping, one-to-one or one-to-many or many-to-many as it doesn't map to a smallest unit but a space
consisting of this smallest unit!

Consider Case 2:

Our petite point is a floating point.

Well floating points are just a representation of the real world fixed point in digital world. So the same argument hold here too.

Q.E.D

I always loved to scribble Q.E.D. My Quite Easily Done attempt.