# The Domain of Hyper-real Functions

We know that the set of all real numbers is a subset of the hyperreal numbers, and the extension principle allows us to apply every real function to hyperreal numbers.

For every real function $f$ of one or more variables we are given a corresponding hyperreal function $f^*$ of the same number of variables. $f^*$ is called the natural extension of $f$.

I'm not sure about the domain of hyperreal functions. For example, here is a real function of one variable:

$f(x)=x^2$,

so the natural extension of this function is $f^*(x)=x^2$, and probably the domain of this function would be a set $D=\{x|x ∈ \mathbb{R}^* - \mathbb{R}\}$. What I've in mind for the natural extension of a real function is that, they can only have for their input the hyperreal numbers which are not real numbers because I think they have especially been formed for those hyperreal numbers that are not reals such as infinitesimals, infinite hyperreal numbers, etc.

What is the possible set of values that can go into a certain hyperreal function as inputs? Is that the whole set $\mathbb{R^*}$ or ($\mathbb{R}^*-\mathbb{R}$)?

• If you start with a function defined on $\mathbb{R}$, you should only call something an "extension" if it is also defined on $\mathbb{R}$. The domain for this function is all of $\mathbb{R}^*$. Oct 11, 2013 at 21:16

More generally, for each real function $f:D\to \mathbb{R}$, its natural extension $f^*$ will have domain $D^*$. Here $D^*$ always contains $D$. If $D$ is an infinite set, then $D^*$ will always contain additional "nonstandard" elements not contained in the original domain $D$. In the ultrapower construction, $D^*$ is represented by the constant sequence $(D,D,D,\ldots)$.