Okay, I promise, this is going to be the last post I write about ultrafunctions, but I just wanted to gather all the information on them in one place.
Definition 1: The ultrafunction is defined as (applied b) times
Theorem 1: f’s range must be the same set as, or a subset of, its domain (that is, )
Proof: Consider . Since f is a function in a single argument a, it only makes sense to apply f to itself if f’s result is contained in its domain. This can naturally, and easily be extended to a larger chain of functions [such as ]. Q.E.D.