Consider a pair of random variables $(X,Y)$ that have a joint density $f_{X,Y}(x,y)$ (wrt Lebesgue or counting measure, or their combination) and a diffeomorphism $T: R^2 \rightarrow R^2$ (differentiable, bijective, and its inverse is also differentiable). Let $(U,V)=T(X,Y)$. If both $X$ and $Y$ are discrete, then $f_{U,V}(u,v)=f_{X,Y}(x(u,v),y(u,v))$ (express $x,y$ in terms of $u,v$); if both $X$ and $Y$ are absolutely continuous, then $f_{U,V}(u,v)=f_{X,Y}(x(u,v),y(u,v))|det(J)|$, where $J$ is the Jacobian of $x,y$ relative to $u,v$.
My question is that what is the formula for $f_{U,V}(u,v)$ when $X$ is discrete but $Y$ is absolutely continuous ($X,Y$ not necessarily independent)?
[Just to give an example of when something like this might occur: in clustering or hierarchical modelling settings, we might have $f_{X,Y}(x,y)= f_{Y|X}(y|x) f_X(x)$ and $f_{Y|X}(y|x)=:g_x(y)$ is a density we are familiar with. (Note this is a special case of the above question).]