How can we answer that, without first having a better understanding of what we mean by "time" - and without first being certain that the various formulations we have of time actually correspond to what we mean?
Let's try a novel approach. Start by characterizing time as the passing of events. A key property of the events is that they have an ordering with respect to one another, some events take place after others and are connected to those that they follow. It's a partial ordering, as far as we know: remotely-situated events and unrelated events do not exhibit any definite before-after relation with one another, although they may be endowed with such a relation in the representations we have.
Historically, the representation that comes to mind is that arising from the Newtonian world. Its main distinguishing feature is that it imposed an ordering on events - no matter how far situated they were from one another - based on when they occurred, as reckoned by a universal clock. Only events that resided in the same 3D snapshot with one another were counted as being neither before nor after, but simultaneous.
The transition from the Newtonian world to the world of Relativity is well-accounted for and well-known, and I won't go in too much detail about it, except to note that the chief premise that there be planes of simultaneity was revoked in such a way as to allow for the existence of events A, B and C such that C is after A, but B is neither before nor after A or C. The chief example would be two ticks of a clock one second apart on the moon (A and C) with B being an event on Earth that lies in a certain small time window of about 1 1/2 seconds (twice the distance to the moon, in light seconds, minus 1 second).
Both of these cases impose an external, universal relation on events of some sort that glazes over the question of what connection (if any) any given pair of events may have to one another. And, neither addresses the question, definitely, of what we actually mean by an "event" - other than to beg the question by calling every single point in the underlying chrono-geometry an "event". It begs the question in two ways: (1) that there actually be any event that happens at the given point, and (2) that only one event can happen there.
So, the novel approach I'd like to try here is to identify an event with that which occurs when a measurement is done. What a measurement is, exactly, is the topic of Measurement Theory, which also happens to be ground zero of the issue of quantum theory interpretations. But it's not resolved.
Also, not definitely resolved is the question of what constitutes a before-after relation for measurements. When can we say that one measurement actually occurs before another - with the specific aim, in mind, that the earlier one should have some actual connection to the later one? And what kind of ordering relation results? A partial ordering? Or do cycles exist in the relation graph? Cycles exist, if time-travel loops exist or other forms of causality violation exist - like those considered by Feynman and Wheeler in their 1949 paper "Classical Electrodynamics in Terms of Direct Interparticle Action" (Reviews of Modern Physics 21 (3), accessible on-line), where they introduced the idea of the "glancing blow" solution to the time-travel paradox ... that was later picked up on and reiterated by Friedman et al. in their early 1990's time travel paper.
Whatever the answer to those questions may be, with this view, time is made out of events, each one being a measurement. So, the question of whether time is discrete or continuous now boils down to the question of whether the before-after ordering of measurement events is discrete or dense. What is its topology? Is it possible to pose a third measurement between any two measurements that are before and after one another, such that all three lie in before-after sequence, or is there a limit to how far this can be taken?
An example of a sequence of measurements are the consecutive ticks on a clock, each one marking some kind of measurement event (namely, the tick, itself, which we assume has some tangible, physical form). So, an upshot of the question is: can we double the resolution and frequency of an already-existing clock? And, can we do so for any clock, no matter when or how conceived? Or, is there a highest frequency?
Before you jump and say "Planck scale!", let me remind you of something. The Planck scale is composed from three physical quantities: Planck's constant h, the in-vacuo speed of light c, and Newton's gravitational coefficient G. But most theoretical physicists ascribe the notion that the 3+1 dimensional gravitational physics described by General Relativity (and described classically by Newtonian gravity), is not fundamental, but is built on a deeper layer of some sort, consisting of a law of gravity in a chrono-geometry of a larger number of dimensions.
That law of gravity has its own version of G, while the G you know of would not be fundamental at all, but derived from the higher-dimensional G and non-fundamental properties, like the size and shape of the higher unseen dimensions (e.g. the volume of a fibre, if the underlying geometry is a principal bundle or homogeneous space with a 3+1 dimensional base space).
The version of the "Planck scale" formed from c, h and the higher-dimensional G may be something entirely different from that formed from c, h and our G. They may not be anywhere near the same size at all. So, to those who say "Planck scale!": that, too, is unresolved, Which "Planck scale"? The one constructed from c, h and higher-dimensional-G or the presumably non-fundamental one constructed from c, h and our G?
And an explanation would still be required to answer the question of what bearing, if any, it has on the ability to interpose measurement events between one another.