If I have a function $f : \mathbb R \to \mathbb R$ and extend it to the hyperreal function $f^* : \mathbb R^* \to \mathbb R^*$, what are some of the properties that I know $f^*$ must have?
Specifically, is there any "defining" property? That is, is there a property $P$ such that we can say
$f^*$ is the only hyperreal function that agrees with $f$ on the reals and satisfies $P$
? (Analogously to how for any uniformly continuous function defined on a dense set, there is exactly one continuous extension defined on all reals)
Edit: Apparently, I didn't make it clear enough what, exactly, I am interested in, and I'm sorry for that. I am not asking which properties of $f$ carry over to $f^*$; I am asking, if you don't know $f$, which properties can still conclude $f^*$ must have.
To rephrase my question in a way that, I hope, eliminates this ambiguity: If you hav some function $g : \mathbb R^* \to \mathbb R^*$, what are some necessary conditions of the proposition
There is some function $f : \mathbb R \to \mathbb R$ such that $g = f^*$
?
The comment under your question pretty much says it all but it may be worth spelling it out in case you are not familiar with logical terminology. Consider the following two examples:
(1) definition of continuity of a function $f$ at $c$: $\forall\epsilon>0\;\exists\delta>0\;\forall x \;(|x-c|<\delta\implies|f(x)-f(c)|<\epsilon)$.
(2) completeness property of the real number field: $\forall S\subseteq\mathbb{R}$ if $S$ is bounded above then there exists a least upper bound for $S$.
I didn't bother writing out the fully formalized definition in (2) because the first quantifier is sufficient to illustrate my point.
The transfer principle applies to statements like (1) which involve quantification over elements only, but it does not apply directly to statements like (2) which involves quantification over sets of elements.
In fact, the completeness property (2) applied literally to the hyperreals would fail. Consider for example the set of all finite hyperreals.
In the comments the OP indicated that he is thinking in terms of the ultrapower construction of $\mathbb{R}^\ast$ relative to an ultrafilter $U$. Then the condition that $g$ should be a natural extension is simply the following. Consider the restriction of $g$ to $\mathbb{R}$ and denote it by $f$. Now apply the construction modulo the same ultrafiter $U$ to the function $f$, getting $f^\ast$. Now $g$ and $f^\ast$ must coincide at every hyperreal point.
A weaker condition is that $g$ must be given by an internal set to begin with.