An intrinsic approximation principle characterizing continuity of Scott function spaces

Let $X$ and $L$ be continuous domains, and let $T$ denote the domain used in the Scott function spaces $[X\to T]$ and $[T\to L]$. The paper considers finite-layer truncation maps and finitely separating approximate identities as approximation mechanisms for such function spaces. Intrinsic approximation principle conjecture. There exists an intrinsic approximation principle for continuous domains that simultaneously generalizes finite-layer truncation maps and finitely separating approximate identities, and whose presence characterizes continuity of the corresponding Scott function spaces. This would unify the sufficient conditions developed in the paper and identify the intrinsic property governing continuity of these Scott function spaces; the principle is not specified or established here.