Consider a point charge q at the vertex of an arbitrary cube. If asked to consider the flux the cube experiences, q/8epsilon seems a natural answer, by constructing seven more such cubes to create an overall cube of 8 times the volume with q at its center. But, this doesn't make sense to me. This seems to imply that every such cube is containing an octadrant of the spherical point charge. But it is a POINT charge. You can't divide it. You can have zero, or you can have q, but nothing in between, when it comes to creating your surface. So, is this an improper gaussian surface ? I am aware that , when it comes to choosing gaussian surfaces, some shapes ( like a Klein bottle ) aren't acceptable. Is this one of them ? More generally, is a discontinuity in p(r), E(r) or V(r) enough to nullify choosing a gaussian ? If I have to use a dirac delta to describe any of them , doesn't that mean my functions are mathematically improper, and that something's off ?