I was replying to your "not typically used in mathematics" quote. As to the "and [thus] are not analogous" part, I recommend the Concrete Mathematics book. You

I was replying to your "not typically used in mathematics" quote. As to the "and [thus] are not analogous" part, I recommend the Concrete Mathematics book. You'll certainly see at least a hundred examples of continuous examples alongside with their discrete counterparts, as you can imagine by looking at the book title. Half-open intervals feature prominently when doing discrete calculus in the Sums chapter.