  We prove a tight uniform continuity bound for Arimoto's version of the
conditional $\alpha$-R\'enyi entropy, for the range $\alpha \in [0, 1)$. This
definition of the conditional R\'enyi entropy is the most natural one among the
multiple forms which exist in the literature, since it satisfies two desirable
properties of a conditional entropy, namely, the fact that conditioning reduces
entropy, and that the associated reduction in uncertainty cannot exceed the
information gained by conditioning. Furthermore, it has found interesting
applications in various information theoretic tasks such as guessing with side
information and sequential decoding. This conditional entropy reduces to the
conditional Shannon entropy in the limit $\alpha \to 1$, and this in turn
allows us to recover the recently obtained tight uniform continuity bound for
the latter from our result. Finally, we apply our result to obtain a tight
uniform continuity bound for the conditional $\alpha$-R\'enyi entropy of a
classical-quantum state, for $\alpha$ in the same range as above. This again
yields the corresponding known bound for the conditional entropy of the state
in the limit $\alpha \to 1$.
