Question

For mass volume density, why is it that when you make delta v go to 0, delta m would vary discontinuously?

Answer 1

is matter a continuous substance, or is it made up of discreet particles...?

Answer 2

To elaborate a bit on TT's excellent hint, consider what happens to delta m as delta v shrinks, and the number of point particles within dv falls. Consider particularly the case when delta v is so small that at most only a handful of particle are within it, and then finally either one or none.