For a univariate distribution or a univariate random variable, we call it continuous/absolutely continuous if its cumulative distribution function (CDF) is continuous/absolutely continuous. Now I am trying to extend this concept to the multivariate case, and want the following holds: The continuity of a random vector does not change under homeomorphisms.

If we still define the continuity of a distribution only according to the continuity of its CDF, then the invariance property above does not hold. This can be seen from the following example.

For any random vector $(X,X)$ where $X$ is continuous, it is easy to check that this vector is continuous (according to the definition above). We transform $(X,X)$ into $(2X,0)$ by the homeomorphism $(x,y)\mapsto (x+y,x-y)$. Obviously, $(2X,0)$ is not continuous.

Now I give two new definitions of the continuity of a random vector.

**(Definition 1) For random vector $(X,Y)$, we call it continuous if $F_{X'Y'}(x,y)$ is jointly continuous in $(x,y)$ for any homeomorphism $(X,Y)\mapsto (X',Y')$. Here $F_{X'Y'}$ is the CDF of $(X',Y')$.**

**(Definition 2) For random vector $(X,Y)$, we call it continuous if $F_X(x)$ is continuous, and $F_{Y|X}(y|x)$ is continuous in $y$ for $P_X$-almost every $x$. Here $F_X$ and $F_{Y|X}$ are the CDF of $X$ and the conditional CDF of $Y$ given $X$, and $P_X$ is the distribution of $X$.**

My question is: For Definition 2, does the continuity of a random vector change under homeomorphisms? If yes, for any continuous random vector $(X,Y)$, is $(Y,X)$ also continuous?