  It is well-known that global hyperbolicity implies that the Lorentzian
distance is finite and continuous. By carefully analysing the causes of
discontinuity of the Lorentzian distance, we show that in most other respects
the finiteness and continuity of the Lorentzian distance is independent of the
causal structure. The proof of these results relies on the properties of a
class of generalised time functions introduced by the authors in
\cite{RennieWhale2016}.
