I want to make sure I understand how to deal with discontinuities when they lie within an interval of integration. Infinite Discontinuities In this case, do we always resort to improper integrals? Jump Discontinuities If there is a finite number of jump discontinuities in the integration interval, we could always use the following property: $$\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$$ Removable Discontinuities We deal with removable discontinuities in a similar way to how we deal with jump discontinuities. What I find confusing, though, is the fact that there are two types of removable discontinuities. The function may or may not be defined at a point of discontinuity. Does this make any difference? Or is the summation rule still valid in this case?