  In this short note, I show how a recent result of Alhejji and Smith
[arXiv:1909.00787] regarding an optimal uniform continuity bound for classical
conditional entropy leads to an optimal uniform continuity bound for quantum
conditional entropy of classical--quantum states. The bound is optimal in the
sense that there always exists a pair of classical--quantum states saturating
the bound, and so no further improvements are possible. An immediate
application is a uniform continuity bound for entanglement of formation that
improves upon the one previously given by Winter in [arXiv:1507.07775]. Two
intriguing open questions are raised regarding other possible uniform
continuity bounds for conditional entropy, one about quantum--classical states
and another about fully quantum bipartite states.
