A random variable in a denominator

Question

For some random variable $X$ & $Y$, I need to calculate the pdf of $Z=\frac{X}{Y}$. I've managed to calculate $$F_Z(t)=\mathbb{P}\left(\frac{X}{Y}\leq t\right)=\mathbb{P}(X\leq t\cdot Y)\,,$$ but I didn't treat to the necessity of $Y\ne 0$. In what way in the answer should I treat this necessity?

Answer 1

If $P(Y = 0) > 0$, then there is a nonzero probability that $Z$ is undefined due to dividing by zero.

However, if $P(Y = 0) = 0$ (for example, if $Y$ has a PDF), then the probability of $Z$ being undefined is zero, so you can essentially ignore that case.