Definition

$X$ and $Y$ are jointly continuous if there exists a non-negative function $$f:\mathbb{R}\times\mathbb{R}\to[0,\infty)$$ such that for any reasonable set $C\subset\mathbb{R}^2$ (measurable), we have $$P((X,Y)\in C)=\int\int_Cf(x,y)dxdy$$RVs $X$ and $Y$ are called jointly continuous and $f$ is their joint pdf.