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The question is exactly the title. Is there a good classification of which functions from $\mathbb{N}$ to $\mathbb{N}$ (or, more generally, from $\mathbb{N}^n$ to $\mathbb{N}$)? Also, what is a good source to learn about $\beta\mathbb{N}$?

I think I can prove that every function extends uniquely, but this seems a little strong, especially since I can’t find this fact anywhere.

(My argument is roughly this: since $\mathbb{N}$ is dense in $\beta\mathbb{N}$, we only need to show extension; uniqueness is immediate. Let $f: \mathbb{N}\rightarrow\mathbb{N}$. Define a new function $\hat{f}: \beta\mathbb{N}\rightarrow\beta\mathbb{N}$ by $\hat{f}(\mathcal{U})=\lbrace X: \exists A\in \mathcal{U}(f(A)\subseteq X)\rbrace$. This extends $f$, and takes values in $\beta\mathbb{N}$, so the only thing to check is that it is continuous. To see this, take some $B\subseteq \mathbb{N}$; we need to show that the $\hat{f}$-preimage of $\lbrace \mathcal{U}: B\in\mathcal{U}\rbrace$ is open. But the preimage of a single ultrafilter is open, so this is clear. As a subquestion, is this argument correct? Or salvagable?)

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You were doing fine until you got to the proof of continuity. For that you want the following calculation, where for $A\subseteq\Bbb N$ I set $\widehat A=\{\mathscr{U}\in\beta\Bbb N:A\in\mathscr{U}\}$.

$$\begin{align*}

\hat f^{-1}\left[\widehat B\right]&=\left\{\mathscr{U}\in\beta\Bbb N:B\in\hat f(\mathscr{U})\right\}\\

&=\left\{\mathscr{U}\in\beta\Bbb N:\exists U\in\mathscr{U}\Big(f[U]\subseteq B\Big)\right\}\\

&=\left\{\mathscr{U}\in\beta\Bbb N:\exists U\in\mathscr{U}\Big(U\subseteq f^{-1}[B]\Big)\right\}\\

&=\left\{\mathscr{U}\in\beta\Bbb N:f^{-1}[B]\in\mathscr{U}\right\}\\

&=\widehat{f^{-1}[B]}

\end{align*}$$

Suppose that $f:\Bbb N\to X$ is a sequence in some compact Hausdorff space $X$, and $\mathscr{U}\in\beta\Bbb N$. For each $U\in\mathscr{U}$ let $K_U=\operatorname{cl}_Xf[U]$, and let $\mathscr{K}=\{K_U:U\in\mathscr{U}\}$; clearly $\mathscr{K}$ is centred, so $\bigcap\mathscr{K}\ne\varnothing$. Let $x\in\bigcap\mathscr{K}$, and let $V$ be an open nbhd of $x$. Let $A=\{n\in\Bbb N:f(n)\in V\}$. If $A\notin\mathscr{U}$, let $U=\Bbb N\setminus A\in\mathscr{U}$; then $V\cap K_U=\varnothing$ a contradiction. Thus, $\{x\in\Bbb N:f(n)\in V\}\in\mathscr{U}$ for each open nbhd $V$ of $x$. Since $X$ is Hausdorff, this easily implies that $\bigcap\mathscr{K}=\{x\}$, and we write $x=\mathscr{U}\text{-}\lim f$.

What you’re doing is letting $\hat f(\mathscr{U})=\mathscr{U}\text{-}\lim f$ for each $f:\Bbb N\to\Bbb N$, taking $X$ to be $\beta\Bbb N$. This construction is one way to prove the general fact that every function $f:\Bbb N$ to a compact Hausdorff space extends to $\beta\Bbb N$. Of course if you know that the Čech-Stone compactification $\beta X$ has (and is characterized among compactifications by) the property that any continuous function from $X$ to a compact Hausdorff space extends to $\beta X$, then you don’t have to go through the explicit calculations. It sounds from your question, though, as if you’re also interested in the nuts and bolts of $\beta\Bbb N$.

**Added:** What would be a good introduction to $\beta\Bbb N$ depends a lot on your interests. Are you interested in it from a primarily set-theoretic point of view, with particular interest in different types of ultrafilters, or is your interest more topological, so that you’re interested in Čech-Stone compactifications generally? For the latter there’s always the classic *Rings of Continuous Functions*, by Gillman & Jerison; it’s dated, but there’s still a lot of good basic material there. A *little* less dated is *The Theory of Ultrafilters*, by Comfort & Negrepontis. Jan van Mill’s chapter, *An introduction to* $\beta\omega$, in the *Handbook of Set-Theoretic Topology*, ed. K. Kunen & J.E. Vaughan, requires a bit of background but is well worth reading once you have the background. My library is a bit old now, and there may well be some good treatments that are more recent than any with which I’m familiar.

There’s a more general principle at work here. Taking Stone–Čech compactifications is functorial: it not only assigns to every topological space $X$ a compact Hausdorff space $\beta X$, it assigns to every continuous map of topological spaces $f : X \to Y$ a continuous map $\beta f : \beta X \to \beta Y$, and it does so in a way that is compatible with composition.

This can be seen in several way depending on exactly how you construct the Stone–Čech compactification. Morally the reason the Stone–Čech compactification is functorial is that it is left adjoint to the inclusion functor from compact Hausdorff spaces to spaces, although proving that such a left adjoint exists requires some work.

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