## The Complete Guide to Ultrafunctions

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 $u_n\left(f\left(x\right)\right)$ is defined as $f\left(f\left(\dots f$$a$$\dots\)$ (applied b) times

Theorem 1: f’s range must be the same set as, or a subset of, its domain (that is, $f:A\rightarrowA$)

Proof: Consider $f(f(a))$. 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 $f(f(f(a)))$]. Q.E.D.