I was coding a physics simulation, and noticed that I was using discrete time. That is, there was an update mechanism advancing the simulation for a fixed amount of time repeatedly, emulating a changing system.

I though that was interesting, and now believe the real world must behave just like my program does. It is actually advancing forward in tiny but discrete time intervals?

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: 
The discrete case is essentially untractable since analysis (the tool created by Newton, in a sense the father of modern physics) can no longer be applied.

Edit: If time appears 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. 
