Measurement takes time; measurement is a process. So the measurement of time immediately yields this theoretical issue:
Since measurement takes time, our ability to break time into ever smaller pieces will always be proportional to the method of measurement used. The faster our measurement device that measures time, the more divisible time will be. Insofar as there are limits to how fast a measurement process can occur (relativistic or other), there will be limits on the lengths of time we can measure. From this perspective, time is discontinuous: there will be a point at which we can no longer split time into smaller pieces.
From a different perspective, time must be continuous: we can start our measurement of time whenever. Since there are no restrictions on when our measurement may begin, each and every instant must be just as good as every other instant, hence time is continuous.
So which is it: Is time continuous or discontinuous?
Or is the question badly formed? The discontinuity argument is based upon the ideas of measurement and relativity. The latter argument, for continuity, is based upon what might be considered a fact of modal reality. Perhaps the two arguments are not talking about the same thing.
I can’t give an end-all be-all answer to the questions of time, but here is my opinion: Time is continuous, but when we start to do scientific activities, time can and will only be able to be measured discretely. Therefore the two arguments are not using one word to describe two different phenomena.
The question then becomes how doing science limits what we can observe.
This might sound like an extremely unlikely situation, but consider the case of organized sports. When playing a sport or game you are bound, restricted, to following certain rules. However, by following these rules, you and the other players can demonstrate skills and abilities that you otherwise would not have been able to observe: Lots of people may be in shape, but only a small fraction of those people are professional athletes. Those athlete demonstrate their superior physical and mental prowess by performing on the game field by being restricted by the official rules.
Getting back to science, does it now seem so unlikely that we restrict ourselves in certain ways in order to accomplish other tasks? For time to be scientifically useful, we need to have some sort process that has a fixed point from which to start counting from, and a unit to count. Then we can compare an unknown process to this known process, and we have done so with much success.
This comparison could not have occurred without the introduction of an arbitrary fixed point and unit of measurement: by restricting our concept of time to these particular processes we enable ourselves to perform scientific research. Research is not possible if we use the unrestricted modal notion: no comparison can be made because there is no inter-modal process to compare a worldly (intra-modal) phenomenon to. But with the use of fixed points, units and processes, we also become subject to relativistic limitations. It seems like a very small price to pay considering the success of science.
To sum up: time is subject to modal considerations, which gives it special properties such as being continuous. Once we start to do science, though, we restrict ourselves to the non-modal aspects of time, which allows us to use it as a tool in scientific research. This also makes time appear to have different properties, but upon closer study, these properties are artifacts of the measurement process and not time itself.