If space-time were not quantized, we would not be able to cover distances and time would not pass. I know about infinitesimal and the Newtonian solution of the zeno's paradox but it strikes me as a mathematical construction of convenience which works due to the physical reality of quantization of space-time.
And From: http://www.askamathematician.com/2013/0
Planck length (or distances of approximately that size- the smallest particle, the electron, is about 10^20 times larger (that’s the difference between a single hair and a large galaxy). ) do show up. In particular, it shows up in the “Generalized Uncertainty Principle” (GUP) where it’s inserted basically as a patch to make physics work in some fairly obscure situations (quantum gravity). The GUP implies that at a small enough scale it is literally impossible, in all situations, to make a smaller-scale measurement. In the right light this makes it look like maybe spacetime is discrete and comes in “smallest units”, and maybe the universe is like the image on a computer screen (made up of pixels). Like a lot of things in string theory (this is an opinion), these sort of patches may prove to be mistakes. So, spacetime may come in discrete chunks, but the most we can say is that those chunks (if they exist) are very, very, very, very small.You’d never notice (at least, the experiments designed to notice haven’t so far). A hydrogen atom is about 10 trillion trillion Planck lengths across.