empirical quantile function - uniform convergence

Question

Let $X_1,...,X_n$ denote independent and identically distributed random variables, with $X_i \sim F$, $1 \leq i \leq n$. Assume $F$ is continuous. Then we know that its generalized inverse (quantile function) $F^{\leftharpoonup}(u):= \inf\{x: \, F(x)>u\}$ exists. If $F_n$ denotes the empirical cumulative distribution function, i.e. $F_n(x)=\frac{1}{n} \sum_{i=1}^n \textbf{1}(X_i \leq x)$, by Glivenko-Cantelli we know $\Vert F_n - F \Vert_\infty \overset{a.s.}{\to}0$. Now, what can we say about $ \Vert {F_n}^{\leftharpoonup}- F^{\leftharpoonup} \Vert_\infty $? Are there conditions under which a.s. uniform convergence can be obtained for the empirical quantile function ${F_n}^{\leftharpoonup}$?