Continuous iff Way Below iff There Exists Element that Way Below and Way Below

Theorem
Let $\left({S, \preceq_1, \tau_1}\right)$ and $\left({T, \preceq_2, \tau_2}\right)$ be complete continuous topological lattices with Scott topologies.

Let $f: S \to T$ be a mapping.

Then $f$ is continuous
 * $\forall x \in S, y \in T: y \ll f\left({x}\right) \iff \exists w \in S: w \ll x \land y \ll f\left({w}\right)$

Sufficient Condition
Assume that
 * $f$ is continuous.

By Continuous iff Directed Suprema Preserving:
 * $f$ preserves directed suprema.

By Directed Suprema Preserving Mapping at Element is Supremum:
 * $\forall x \in S: f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$

Thus by Mapping at Element is Supremum implies Way Below iff There Exists Element that Way Below and Way Below:
 * $\forall x \in S, y \in T: y \ll f\left({x}\right) \iff \exists w \in S: w \ll x \land y \ll f\left({w}\right)$

Necessary Condition
Assume that
 * $\forall x \in S, y \in T: y \ll f\left({x}\right) \iff \exists w \in S: w \ll x \land y \ll f\left({w}\right)$