A gentle introduction to information geometry

Why it has to be continuous? I know it is important and wider coverage. But can it be N^n instead? Those point can be Neigbour as well for f and f-1. Discrete information space is important to well this is HN and computer oriented.

I am also working on Zeno ... and I argue that discrete information space cannot solve so is continuous.

My ultimate is 2 dim with only 1 dim is continuous and maps to [0,1] like qm (as imaginary number or a real number orthogonal to the other real number).