  We completely classify the orientable infinite-type surfaces $S$ such that
$\operatorname{PMap}(S)$, the pure mapping class group, has automatic
continuity. This classification includes surfaces with noncompact boundary. In
the case of surfaces with finitely many ends and no noncompact boundary
components, we prove the mapping class group $\operatorname{Map}(S)$ does not
have automatic continuity. We also completely classify the surfaces such that
$\overline{\operatorname{PMap}_c(S)}$, the subgroup of the pure mapping class
group composed of elements with representatives that can be approximated by
compactly supported homeomorphisms, has automatic continuity. In some cases
when $\overline{\operatorname{PMap}_c(S)}$ has automatic continuity, we show
any homomorphism from $\overline{\operatorname{PMap}_c(S)}$ to a countable
group is trivial.
