Mapping Preserves Infima implies Mapping is Continuous in Lower Topological Lattice

Theorem
Let $T = \left({S, \preceq, \tau}\right)$ and $Q = \left({X, \preceq', \tau'}\right)$ be complete topological lattices with lower topologies.

Let $f: S \to X$ be a mapping such that
 * $f$ preserves all infima.

Then $f$ is continuous mapping.

Proof
By assumption:
 * for all non-empty subsets $Y$ of $S$: $f$ preserves the infimum of $Y$.

Thus by Mapping Preserves Non-Empty Infima implies Mapping is Continuous in Lower Topological Lattice:
 * $f$ is continuous mapping.