Let $(X,Y)$ be a random vector with CDF $F$. In general, continuity of the marginal distributions of $X$ and $Y$ does not allow one to conclude that $F$ is continuous. Under what conditions can we conclude that $F$ is continuous as well --- I think independence of $X$ and $Y$ is a special case under which this holds true, but are there other special cases?
The reason I am asking this is because I came across the following: we have a sequence of i.i.d random vectors $\{(X_i, Y_i)\}$ and we suppose there exists a normalization such that $$\lim_{n\to \infty} P(\frac{\max(X_1,...,X_n)-b_n}{a_n}\leq x,\frac{\max(Y_1,...,Y_n)-d_n}{c_n}\leq y)=G(x,y)$$ for all continuity points of the distribution function $G$ with nondegenerate marginals. Then, the author shows that the marginals must be continuous and concludes that $G$ must be continuous as well. Why is this true?
(this is from page 209 of "Extreme Value Theory, De Haan & Ferreira")