As far as I am aware, if $X$ and $Y$ were jointly continuous, then this implies that $X$ and $Y$ are individually continuous random variables, with densities which can derived from the joint density of $X$ & $Y$ (by computing the marginals).
Additionally, I am aware that the converse is not always true.
However, I have seen proofs in probability textbooks that state that $X$ and $Y$ are individually continuous, and then proceed to talk about the joint density of $X$ and $Y$, which implies that they are jointly continuous which I thought wasn't always necessarily true?
Is it fair to conclude that if $X$ and $Y$ are individually continuous, then they are always jointly continuous? If so, why?