=== Sufficient Condition === Assume that :$f$ is continuous. Let $\left({D, \precsim}\right)$ be a directed set. Let $N:D \to S$ be a Moore-Smith sequence in $S$. Aiming for a contradiction suppose that :$f\left({\liminf N}\right) \npreceq_2 \liminf\left({f \circ N}\right)$ By definition of lower closure of element: :$f\left({\liminf N}\right) \notin \left({ \liminf\left({f \circ N}\right) }\right)^{\preceq_2}$ By definition of relative complement: :$f\left({\liminf N}\right) \in \complement_T\left({\left({ \liminf\left({f \circ N}\right) }\right)^{\preceq_2} }\right)$ By Lower Closure of Element is Topologically Closed in Scott Topological Ordered Set: :$\left({ \liminf\left({f \circ N}\right) }\right)^{\preceq_2}$ is closed. By definition of closed set: :$\complement_T\left({\left({ \liminf\left({f \circ N}\right) }\right)^{\preceq_2} }\right)$ is open. By definition of continuous: :$f^{-1}\left[{\complement_T\left({\left({ \liminf\left({f \circ N}\right) }\right)^{\preceq_2} }\right)}\right]$ is open. By Open iff Upper and with Property (S) in Scott Topological Lattice: :$f^{-1}\left[{\complement_T\left({\left({ \liminf\left({f \circ N}\right) }\right)^{\preceq_2} }\right)}\right]$ has property (S). By Set of Infima for Sequence is Directed: :$X := \left\{ {\inf \left({N\left[{\precsim\left({j}\right)}\right]}\right): j \in D}\right\}$ is directed. By definition of limit inferior: :$\liminf N = \sup X$ By definition of preimage of set: :$\liminf N \in f^{-1}\left[{\complement_T\left({\left({ \liminf\left({f \circ N}\right) }\right)^{\preceq_2} }\right)}\right]$ By definition of property (S): :$\exists y \in X:\forall x \in X: y \preceq_1 x \implies x \in f^{-1}\left[{\complement_T\left({\left({ \liminf\left({f \circ N}\right) }\right)^{\preceq_2} }\right)}\right]$ We have: :$\exists j \in D: y = \inf \left({N\left[{\precsim\left({j}\right)}\right]}\right)$ By definition of reflexivity: :$y \preceq_1 y$ Then: :$y \in f^{-1}\left[{\complement_T\left({\left({ \liminf\left({f \circ N}\right) }\right)^{\preceq_2} }\right)}\right]$ Define $y' := \inf \left({\left({f \circ N}\right)\left[{\precsim\left({j}\right)}\right]}\right)$ Define $X' := \left\{ {\inf \left({\left({f \circ N}\right)\left[{\precsim\left({j}\right)}\right]}\right): j \in D}\right\}$ By Mapping is Increasing implies Mapping at Infimum for Sequence Precedes Infimum for Composition of Mapping and Sequence: :$f\left({y}\right) \preceq_2 y'$ By definition of $X'$: :$y' \in X'$ By definitions of supremum and upper bound: :$y' \preceq_2 \sup X'$ By definition of limit inferior: :$\liminf \left({f \circ N}\right) = \sup X'$ By definition of transitivity: :$f\left({y}\right) \preceq_2 \liminf \left({f \circ N}\right)$ By definition of lower closure of element: :$f\left({y}\right) \in \left({\liminf \left({f \circ N}\right)}\right)^{\preceq_2}$ By definition of preimage of set: :$y \in f^{-1}\left[{\left({\liminf \left({f \circ N}\right)}\right)^{\preceq_2} }\right]$ By Complement of Preimage equals Preimage of Complement: :$f^{-1}\left[{\complement_T\left({\left({ \liminf\left({f \circ N}\right) }\right)^{\preceq_2} }\right)}\right] = \complement_S\left({f^{-1}\left[{\left({ \liminf\left({f \circ N}\right) }\right)^{\preceq_2} }\right]}\right)$ Thus this contradicts $y \in \complement_S\left({f^{-1}\left[{\left({ \liminf\left({f \circ N}\right) }\right)^{\preceq_2} }\right]}\right)$ {{qed|lemma}} === Necessary Condition === Assume that :for all directed set $\left({D, \precsim}\right)$ and Moore-Smith sequence $N:D \to S$ in $S$: $f\left({\liminf N}\right) \preceq_2 \liminf\left({f \circ N}\right)$ By Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Mapping Preserves Directed Suprema: :$f$ preserves directed suprema. Thus by Mapping Preserves Directed Suprema implies Mapping is Continuous: :$f$ is continuous. {{qed}}