Continuous iff Directed Suprema Preserving

Theorem
Let $\struct {S, \preceq_1, \tau_1}$ and $\struct {T, \preceq_2, \tau_2}$ be complete topological lattices with Scott topologies.

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

Then $f$ is continuous $f$ preserves directed suprema.

Sufficient Condition
Assume that
 * $f$ is continuous.

By Continuous iff Mapping at Limit Inferior Precedes Limit Inferior of Composition of Mapping and Sequence:
 * for all directed set $\struct {D, \precsim}$ and net $N:D \to S$ in $S$: $\map f {\liminf N} \preceq_2 \map \liminf {f \circ N}$

Thus by Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Mapping Preserves Directed Suprema:
 * $f$ preserves directed suprema.

Necessary Condition
Thus by Mapping Preserves Directed Suprema implies Mapping is Continuous:
 * if $f$ preserves directed suprema, then $f$ is continuous.