A function needs to be continuous in order to be integrable.
However, can single point removable discontinuities be ignored while integrating? I ask this because obviously a point isn't going to add anything to the area of the graph which is what integration returns. Is it also the case with improper integrals of discontinuous functions. I feel it shouldn't be valid for them because "too many points" finally might end up resulting in a considerable area.
Edit: Referring to Reimann Integrability.