Is there a discontinuous random vector with continuous components?

Question

Is it possible that $X$ and $Y$ are continuous random variables (i.e., their cdfs are continuous), yet the random vector $(X,Y)$ is discontinuous (i.e., their joint cdf is discontinuous) ?

Answer 1

No. We have $(A \cap B) \setminus (C\cap D) \subseteq (A \setminus C) \cup (B \setminus D)$. Take $A=(X \leq x),B=(Y \leq y), C=(X \leq x_n), D=(Y\leq Y_n)$ where $x _n$ increases to $x$ and $y_n$ increases to $y$. Conclude that $P( X \leq x_n, Y\leq Y_n)) \to P( X \leq x, Y\leq Y))$