  We study functions on topometric spaces which are both (metrically) Lipschitz
and (topologically) continuous, using them in contexts where, in classical
topology, ordinary continuous functions are used. We study the relations of
such functions with topometric versions of classical separation axioms, namely,
normality and complete regularity, as well as with completions of topometric
spaces. We also recover a compact topometric space $X$ from the lattice of
continuous $1$-Lipschitz functions on $X$, in analogy with the recovery of a
compact topological space $X$ from the structure of (real or complex) functions
on $X$.

Logic