We completely classify the orientable infinite-type surfaces S such that 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 Map(S) does not have automatic continuity. We also completely classify the surfaces such that 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 PMap_c(S) has automatic continuity, we show any homomorphism from PMap_c(S) to a countable group is trivial.

The author acknowledges support from NSF grant RTG DMS--1745583.