Is time continuous or discrete?

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.

I think it's important to note that quantum or quantized time is not equal to discrete time. For instance, we have "quantized" space. By this we mean that it receives quantum treatment. But the underlying coordinates still form a continuum. So even if you live on a finite circle and only consider wavefunctions so that you get a countable set of basis functions from which to form all the others, you can still in principle measure incidence of particles at any point, again forming a continuum. Therefore, if we take quantum time in analogy to quantum space, we would have to conclude that quantum mechanically it would still form a continuum.

Of course none of this proves how the universe really works, which is your question. The only honest answer direct to your question is "We don't know". Physical theories do not describe how the universe actually works, the only thing we know is that their predictions match experimental results we currently posses. So even if the best physical theories we currently posses use a continuum of temporal coordinates, we cannot by any means conclude that the way the universe actually works matches our description.

The answer to this question is not known presently. Current physics is, as stated by other answers, based on fully continuous mathematical models, which particularly assume spacetime to be continuous. On the other hand you could argue that these models are isomorphic to discrete constructive models, with the general view that the continuous is the limit of the discrete. Some modern spacetime theories assume an underlying network/relational structure, and are fully discrete.

My personal belief is that continuous structures do not exist in the physical world. This is however just a belief.

See also: Is the universe finite and discrete?