Let $P(R)$ be the probability measures on the real numbers $R$ and fix $\alpha \in (0,1)$. Define
$$Q_{\alpha} : P(R) \to R $$
as the function taking a measure $\mu \in P(R)$ to its $\alpha$-th quantile, defined by
$$Q_{\alpha}(\mu):=\inf \{x\in R: \alpha \leq \mu ((-\infty, x ] ) \}.$$
I am intereseted in any results concerning the continuity of the function $Q_{\alpha}$.
I am open as to the distance put on the space $P(R)$ and possibly restricting to probability measures with moments etc.