Prior work on finding domain-theoretic suprema of equivalent total functions?
My question is this: If f is a Scott-continuous computable function, and is &quot;total&quot; (i.e. the elements of its domain are all finite, and it takes &quot;fully-defined&quot; values to &quot;fully-defined&quot; values), have techniques been developed for computing a maximal h such that f⊑h?
Of course, if the domain of f is a finite set, one can just represent f as a lookup table and then use elementary deduction to compute h. It's not necessarily efficient in practice, but it yields the correct answer. But what about functions over infinite domains?

Scott-continuity rules out the lookup table approach. For a concrete counterexample, let's start with the Sierpinksi domain $O = \{\bot, \top\}$, with order $\bot \sqsubseteq \top$.
Intuitively, think of $\bot$ as nontermination, and $\top$ as termination. In Haskell terms, think of this as the domain interpreting expressions of type $()$ -- i.e., you either get back a unit or the expression fails to terminate. We'll also use $2$, the booleans, and $2_\bot$, the pointed booleans.
Now, suppose you have $f : O \to 2_\bot$. You can't continuously complete this function, even though there are only a finite number of possible $f$'s! The intuitive reason is that when you run $f(\bot)$, there's no finite waiting period sufficiently long enough to decide that its return value is bottom -- if there were, then I could give you a function that spins for one second longer, and then returns the opposite of whatever you completed the function to. (Prove that any continuous $h : (O \to 2_\bot) \to (O \to 2)$ must return a constant function independent of its argument $f$.)
Don't think of bottom as an extra element of the domain of values. This will lead you astray. Instead, think of elements of the domain as observations of a computation. Really do think of sitting in front of a terminal, waiting for your program to print something. Bottom represents your observation when the program is still running and hasn't printed anything to the screen -- it represents observing nothing.
The best book I know for developing the right intuitions is Steve Vickers' Topology via Logic. It's a short and easy read, but your domain-fu will go way up for reading it.