This question is a wonderful can of worms. It's the debate between the Eleatics and the Atomists, but with time and not space.

I'll merely point out St. Augustine's wonderful discussion of time from Book XI of his Confessions.

We can and should also consider Husserl's incredible analysis of time in the Phenomenology of Internal Time-Consciousness. These were lectures and notes from around 1905, before Husserl's publication of Ideas I in 1913.

In it, Husserl claims that our experience of time is not merely an experience of the present instant. There is, in fact, something he call the "now phase," but this is not the whole story. The "now phase" will immediately recede into the "just past," like how the comet is followed by its tail tapering off. As the "now phase" recedes into the "just past," a new "now phase" wells up in our experience. Lets think of a melody, a temporal object par excellence:

When the new note sounds, the one just preceding it does not disappear without a trace; otherwise, we should be incapable of observing the relations between the notes which follow one another. We should have a note at every instant, and possibly in the interval between the sounding of the next an empty [leere] phase, but never the idea [Vorstellung] of a melody. (Phenomenology of Internal Time-Consciousness, section 3).

If the past were purely past, incompatible with what is given in the present, then it would not be possible to say, for example, that this pitch is a minor seventh in relation to the previously heard pitch, or that this ii chord was tonicized by the previously heard modified VI chord. The very idea of a melody, a temporal sequence of pitches, would not be known to our consciousness, and we would have no way of distinguishing between a melody and a non-melodic sequence of pitches, like a random smashing of keys on a piano or the confluence of isolated environmental sounds (somebody’s watch beeps, a phone rings, and then a car horn sounds). So, the perception of an object that has gone into the past must somehow still persist in consciousness. But this is not to say that the past is really still the present in an unmodified form, not to grant an honorary status of being to the perceptions that are no longer apprehended by the consciousness. If that were the case, then instead of a melody, a sequence of pitches, we would hear the simultaneous sounding of the melody notes. To the first note would be added the second, the third, and so on until by the end of the melody a dense and cacophonous cluster of pitches would be given to consciousness.

So what Husserl suggests is that part of consciousness' intentional constitution of experience involves a temporal schema that not only apprehends the "now phase," but also is able to "hold on" to the "just past," which Husserl labels retention. Analogously, there are also protentions, which are the intentional rays of consciousness that reach into the future, into what is "about-to-be-now."

So the instant is an artifact of consciousness, a byproduct. It results from the fact that we tend to objectivate time by considering it like a clock - a linear, unidirectional, sequential/serial progression; a metronome. But, Husserl says, at a much deeper level of consciousness our experience of time is constituted by this retention-now phase-protention structure. And what lies beneath that? The answer is wonderfully suggestive: absolute time-constituting consciousness, which Husserl likens to a "flux," and for which

all names are lacking.

Groovy stuff!