My stats course just taught me that a discrete random variable has a finite number of options ... I hadn't realized that. I would have thought, like a set of integers, it could be infinite. Googling and checking a several web pages, including a few from university courses, has failed to specifically confirm this; most sites do however say discrete random variables are countable - I suppose that means finitely numbered?
It is clear that continuous random variables are infinite even if (most?) often bounded.
But if discrete random variables have finite possibilities, what then is an infinite distribution of integers? It's neither discrete nor continuous? Is the question moot because variables either tend to be continuous & (by definition) infinite or discontinuous & finite?