I'm sure this is elementary, but:
If a 1-D function is bounded everywhere except at a point P (measure zero), then the integral exists. So why should it matter if P is at the end of the integration interval?
Presumably the same is true of a line integral in, say, R3 --- which would seem to imply that the path for a closed line integral can include a countable number of discontinuities. But what happens if the closed integral is calculated in such a way that the endpoints a=b are where the function is unbounded?
Taking this one more step, the closed line integral, by Stokes' theorem, can be written as a surface integral of the curl. As long as the regions is simply connected. But if, say, simply connectedness fails at the origin, why should it matter -- isn't that just a set of measure zero for the surface integral?
Clearly, I'm confused and/or missing something basic.
thanks!