Let $x=(a_1, a_2, a_3, ...) + \mathcal U \in {}^\ast \mathbb R := \displaystyle\prod_1^\infty \mathbb R/\mathcal U$ be a hyperreal number using the ultrapower construction and $f \colon \mathbb R\to \mathbb R$ a real function $\mathbb R \to \mathbb R$. We can extend $f$ via the transfer principle to ${}^\ast f \colon {} ^\ast\mathbb R \to {} ^\ast \mathbb R$. Also, let ${}^\ast x:= {} ^\ast f(x)$.
Will we have ${}^\ast x=(f(a_1), f(a_2), f(a_3), ...) + \mathcal U \in {}^\ast \mathbb R$? That is, does ${}^\ast$ "respect" applying $f$ pointwise to every coordinate of ${}^\ast x$?
I don't see how this immediately follows from the transfer principle.