If there are two continuous random variables $X$ and $Y$. What are the additional conditions required on $X$ and $Y$ under which ratio of these random variables $\frac{X}{Y}$ is also a continuous random variable?
I know that divide by zero events should have zero probability and Y should not be scaled version of $X$, in which case ratio is constant. I tried searching real analysis books. Sum and composition of two random variables is clear to me but I am still confused about ratio.
What are the other conditions these two random variables should satisfy such that $\frac{X}{Y}$ is also a continuous random variable?