  In this note, we study F-purity of pairs, and show (as is the case with log
canonicity) that F-purity is preserved at the F-pure threshold. We also
characterize when F-purity is equivalent to sharp F-purity, an alternate notion
of purity for pairs introduced in [Sch08]. These results on purity at the
threshold generalize results appearing in [Har06, Sch08], and were expected to
hold by many experts in the field. We conclude by extending results in [BMS09]
on the set of all F-pure thresholds to the most general setting.
