  We prove a tight uniform continuity bound for a family of entropies which
includes the von Neumann entropy, the Tsallis entropy and the $\alpha$-R\'enyi
entropy, $S_\alpha$, for $\alpha\in (0,1)$. We establish necessary and
sufficient conditions for equality in the continuity bound and prove that these
conditions are the same for every member of the family. Our result builds on
recent work in which we constructed a state which was majorized by every state
in a neighbourhood ($\varepsilon$-ball) of a given state, and thus was the
minimal state in majorization order in the $\varepsilon$-ball. This minimal
state satisfies a particular semigroup property, which we exploit to prove our
bound.
