Product of a discrete and absolutely continuous random variable

Question

I have the independent random variables $U \sim e(1)$ and $V \sim po(2)$. We define the variable $Y=UV$. I want to prove that Y is not absolutely continuous. The only hint I have recieved is that I can use the property that for an absolutely continuous random variable, $P(X=x)=0, \forall x$. However, this doesn't help me when I have no idea how the product of these variables work! Only one of them has a probability mass function, and only one of them has a probability density function.

Answer 1

Hint: $$ \{Y=0\}\supseteq\{V=0\}. $$