If we take the general relativity point of view and understand gravity as curvature of space-time itself, and if we also accept the Heisenberg uncertainty principle, which says that we can never know both the momentum and position of a particle exactly, we must accept that space-time behaves in a quantum manner, because otherwise we could (in principle) measure things arbitrarily accurately with the gravitational field. So theories of quantum gravity seem to say that time itself behaves quantum mechanically at small scales, and that there is a smallest possible time that can _exist_. Nobody claims to understand this very well yet.