As we cannot resolve arbitrarily small time intervals, what is ''really'' the case cannot be decided.
But in classical and quantum mechanics (i.e., in most of physics), time is treated as continuous.
Physics would become very awkward if expressed in terms of a discrete time.
Edit: If time appear discrete (or continuous) at some level, it could still be continuous (or discrete) at higher resolution. This is due to general reasons that have nothing to do with time per se. I explain it by analogy: For example, line spectra look discrete, but upon higher resolution one sees that they have a line width with a physical meaning.
Thus one cannot definitely resolve the question with finitely many observations of finite accuracy, no matter how contrived the experiment.