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__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?