While working through Barendreght's book on the Lambda Calculus, and Abramsky's notes on Domain Theory, I had the following realization:
It's often stated that Domain Theory provides a semantics for the untyped Lambda Calculus, and various other recursive constructions in computer science, by providing canonical (ie, minimal) fixpoints to domain equations like $X\cong [X\to X]_{\bot}$ (This is from "The Lazy Lambda Calculus", by Abramsky(1987), because the canonical fixpoint for $X\cong[X\to X]$ is the single-point set).
Now, one half of the isomorphism makes sense. Obviously every lambda-term either reduces to some suitable notion of normal form (giving rise to a function from lambda-terms to lambda-terms), or it doesn't reduce and can be identified with $\bot$.
However, I'm unsure about the second half of the isomorphism. Despite my reading so far, I still haven't found anything anywhere that has shown that every element in the space of Scott-continuous functions $[X\to X]$ corresponds to a lambda-term. It still looks possible (to me) that the lambda-terms form a subset of $[X\to X]$.
And generalizing further, if you have some domain like the lazy natural numbers (call it $N$), do all the elements of the function-space $[N\to N]$ correspond to a computable function? What conditions on domains $A$ and $B$ are sufficient (or necessary) for all elements of $[A\to B]$ to be represented as a computable function, and is there a standard reference for that information?