User:Vel!/pu/continuous

A work of continuous googology

It hasn't exactly became a craze yet (I seem to be the second one to do this), but continuous googology seems to be getting a lot of attention now. PsiCubed2 made this letter notation that is able to express large numbers continuously - a fast-growing function that can take non-integer arguments and from those output nonintegers. Here is my attempt. It will be FGH-based, but with arrays instead of ordinals.

Up to f0,1
This will simply be a generalization of integer-indexed FGH to nonintegers. f0 can easily be defined for nonintegers. The first step will be to do nonintegral function powers. f0m(n) is equal to n+m, so we can extrapolate that definition to arbitrary functions:


 * f0(n) = n (f is any function)
 * fk(n) = f(fk-1(n)) if k ≥ 1
 * fk(n) = f(n+k*(f(n)-n)) if 0 < k < 1

Good. Now we can define fk(n)


 * f0(n) = n+1
 * fk(n) = fk-1n(n) if k ≥ 1
 * fk(n) = n+k*(f(n)-(n+1))+1 if 0 < k < 1

This is all continuous. So there we have it! Nonintegral sub-omega FGH!