To bring up one point out of any number of points one could: the velocity of water through a tube is roughly log(x), where x is the distance (length of the radius) away from the centre of the tube. So at x=0, i.e., at the exact center of the tube, the velocity of the water flowing through the tube is log(0), or negative infinity.

The problem with blanket statements like "the universe is discrete" is that nature doesn't give a hang what we would "prefer". She is what she is, irrespective of wether we like it or not. Spacetime may or may not be infinite. I suspect so, but even if not, there are infinities everywhere you look. Nature likes her infinities, even if you don't.