  We derive a measurement-independent asymptotic continuity bound on the
observational entropy for general POVM measurements, making essential use of
its property of bounded concavity. The same insight is used to obtain
continuity bounds for other entropic quantities, including the measured
relative entropy distance to a convex a set of states under a general set of
measurements. As a special case, we define and study conditional observational
entropy, which is an observational entropy in one (measured) subsystem
conditioned on the quantum state in another (unmeasured) subsystem. We also
study continuity of relative entropy with respect to a jointly applied channel,
finding that observational entropy is uniformly continuous as a function of the
measurement. But we show by means of an example that this continuity under
measurements cannot have the form of a concrete asymptotic bound.
