The plug-in optimal allocation is continuous at a unique best arm

Fix a rule $\mathrm{choice}$ assigning to every Gaussian parameter vector with a unique best arm a full-support optimal allocation. Then along any sequence of estimates $\hat\mu(s)\to\mu$, where $\mu$ has a strictly best arm,

No continuity of the rule is assumed, and none could be: a rule is only required to select an optimal allocation, and a selection is in general discontinuous. The point is that here there is nothing to select. By uniqueness of the optimal allocation, the rule is pinned down at every parameter with a strictly best arm, so it coincides there with the canonical selection and inherits its regularity.

That regularity is the argmax half of Berge's maximum theorem at a point of unique maximisation: for a jointly continuous objective on a compact feasible set, every maximiser at a nearby parameter is near the unique maximiser at the limit. Continuity of the objective is needed only along the feasible set, which matters because the pair weight $\alpha_i\alpha_j/(\alpha_i+\alpha_j)$ is continuous on the closed simplex but not off it.

The statement is about a sequence rather than a neighbourhood because the estimates fed to the rule need not have a strictly best arm at every round --- only eventually, since having a strictly best arm is an open condition. The rule may return anything at parameters with ties, and the conclusion is unaffected.

This is the hypothesis a tracking sampling rule requires: the targets it follows converge, so its empirical allocation inherits the limit by a Ces`aro argument.