If the distribution of $X+Y$ is discrete then both $X$ and $Y$ are discrete as well

Question

"If the distribution of $X+Y$ is discrete then both $X$ and $Y$ are discrete as well"

I think this is true but I'm not sure how to go about proving it. I thought about using characteristic functions but I only know that since $X+Y$ is discrete, we must have $|\varphi_{X+Y}(t)|\leq 1$. Any help is appreciated, thank you.

Answer 1

What about when $X$ is a continuous distribution and $Y=-X$ or $Y=-X+1$. Then the rv $X+Y$ takes all it's value in the countable set $\{1\}$ with probability $1$.