TITLE: Is there a two-dimensional random variable whose CDF is continuous, but whose marginal CDFs are discontinuous?
QUESTION [7 upvotes]: Is there a two-dimensional random variable $(X,Y)$, whose CDF is continuous, but whose marginal CDFs are discontinuous?
My question differs from this one in that a random variable that has a continuous CDF, which is the condition stipulated in my question, is not necessarily a continuous random variable, which is the condition stipulated in the other question. Thus, my question is more general than the other one.

REPLY [4 votes]: The answer is negative even for coordinatewise continuity.
Let $F_X$ be discontinuous in $x_0$. Then $P(X=x_0)>0$, and, thanks to the continuity of probability, $P(X=x_0,Y\le y)>0$ for some $y\in \mathbb{R}$. However, the continuity of $F_{X,Y}$ in $x$ implies
$$
P(X=x_0,Y\le y) = \lim_{x\to x_0-}P(X\in(x_0,x],Y\le y)\\
=\lim_{x\to x_0-}\big(F_{X,Y}(x_0,y)-F_{X,Y}(x,y)\big)  =0,
$$
which is a contradiction.