Some continuity properties of quantum Rényi divergences

In the problem of binary quantum channel discrimination with product inputs, the supremum of all type II error exponents for which the optimal type I errors go to zero is equal to the Umegaki channel relative entropy, while the infimum of all type II error exponents for which the optimal type I errors go to one is equal to the infimum of the sandwiched channel R\'enyi $\alpha$-divergences over all $\alpha>1$. We prove the equality of these two threshold values (and therefore the strong converse property for this problem) using a minimax argument based on a newly established continuity property of the sandwiched R\'enyi divergences. Motivated by this, we give a detailed analysis of the continuity properties of various other quantum (channel) R\'enyi divergences, which may be of independent interest.