Consider a random vector $(X,Y)$. Denote by $F_{(X,Y)}(x,y)$ its distribution function and by $f_{(X,Y)}(x,y)$ its density function (if it exists). The definition of $f_{(X,Y)}$ varies depending on $(X,Y)$:
Case $(X,Y)$ is a discrete random vector: then $f_{(X,Y)}$ is the function living in $[0,1]$ such that $$F_{(X,Y)}(x,y)=\sum_{m:\,m\leq x}\sum_{n:\,n\leq y}f_{(X,Y)}(m,n).$$
Case $(X,Y)$ is an absolutely continuous random vector: then $f_{(X,Y)}$ is the function living in $[0,\infty)$ such that $$F_{(X,Y)}(x,y)=\int_{-\infty}^x\int_{-\infty}^y f_{(X,Y)}(u,v)\,dv\,du.$$
What happens when $X$ is absolutely continuous and $Y$ is discrete? For me, in this case, the density function $f$ should be the function satisfying $$ F_{(X,Y)}(x,y)=\sum_{m:\,m\leq y}\int_{-\infty}^x f(t,m)\,dt.$$ Is this correct? If it is, is it true that $X$ and $Y$ are independent if and only if $f_{(X,Y)}(x,y)=f_X(x)f_Y(y)$?