I'm not sure that discontinuity at zero (or infinity, or whatever limit) is something that shouldn't be alarming. Ruling out such discontinuities is a common requirement for various things I've seen. But even if we allow it, that doesn't really answer my quandary. I'm not (at the moment) worried about how to define integration; I'm worried about why we even need the notion at all. What is it that forces us away from just doing a simple summation?