Let $^*$-polynomials be defined as hyperfinite polynomials over the hyperreals, i.e. elements of the set $\{ p\in \mathbb{R^R}\mid \exists\big( a:\{0,..,n\}\to\mathbb{R}\space \big)\forall x\in \mathbb{R}:p(x) = \sum_{i=0}^n a(i)x^i\text \}$

Using strong enlargements $^*$, the following statements are true:

Let $f:\mathbb{R} \to \mathbb{R}$ be a function. $$\begin{align} &(i) &\text{There exists a } ^* \text{-polynomial } p \text{ with } f(x)\approx p(x) , x\in\mathbb{R} \\ &(ii) &f \text{ continuous} \Leftrightarrow \text{There exists a } ^* \text{-polynomial } p \text{ with } f(x)\approx p(x) , x\in\mathbb{fin(^*R)} \end{align}$$

Given these two statements, there should be some interesting or easy conclusions to make. Unluckily, my resources didn't give any examples whatsoever, so I feel like I'm missing a starting point to build on about that topic.

What are elementary conclusions that can be deduced using these two statements?


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