Functions $f:\mathbb{N}\to \mathbb{N}$ are easily extended to corresponding functions $f:\mathscr{P\omega\to P\omega}$ by $\forall x\in\mathscr{P\omega}:\ f(x)=\bigcup\limits_{i\in x}f(i)$. But, of course, most functions $f:\mathscr{P\omega\to P\omega}$ can't be constructed by such an "extension". Does the subset of $\mathscr{P\omega\to P\omega}$ functions which can be so constructed comprise a subdomain of $\mathscr{P\omega}$?
That is, since $\mathscr{P\omega}\sim\left[\mathscr{P\omega\to P\omega}\right]$ (where $\left[\ \cdot\ \right]$ denotes the Scott-continuous functions), each such $f\in\left[\mathscr{P\omega\to P\omega}\right]$ can be represented by a corresponding $x_f\in\mathscr{P\omega}$. And I'm asking if the collection of these $x_f$'s for our "extended" functions comprise a subdomain of $\mathscr{P\omega}$. And what else, if anything, interesting can be said about them, i.e., what other properties does this collection possess?
Just some textbook/literature references okay, if this is a standard discussion. I'm not well-versed enough in domain theory to know it offhand, and couldn't google any reference. But it sounds straightforward enough that I'd guess it's been discussed.