The notion of a continuum is a human idea, albeit a very powerful one. Modern mathematics (e.g. Calculus, the mathematics of motion and change) hinges upon it. But like all ideas in mathematics, whether it is an "invention" or "discovery" is a matter of debate, or perhaps a matter of perspective... And regardless, it is a leap of faith to infer that physical space or time (or spacetime) is an example of a continuum, or a manifestation of this idea.
The ancient Greek philosopher Zeno famously contrived a set of paradoxes which demonstrate, from various perspectives, that if space and time are taken to be continuous then motion is logically impossible, by virtue of requiring infinitely many "steps" to be completed in a finite time. ( https://en.wikipedia.org/wiki/Zeno%27s_paradoxes ) Obviously, it is possible to sweep these objections under the rug and just run with it (this is basically how Calculus was developed, "it works, we'll figure out the theory later"), but IMO paradox should never be ignored. It's an indication of conflicting assumptions, that's one's view is not self-consistent.
In modern quantum physics, spacetime is described as a "quantum foam" of unpredictable energy fluctuation, more like static noise than like blankness. And it has a granularity to it in at least some sense: there is a smallest interval of time (and also of space) over which any meaningful physical measurement can be made, called the Planck time (and correspondingly the Planck length for space). To what extent space or time "exist" on smaller scales than this is a very open question in physics circles.
Personally, I do not believe in existence apart from awareness, and so I certainly don't believe in the existence of things that are not measurable even in theory. And so I do not believe spacetime is a continuum. As Padma said, I think a sequence of moments is a better description. In contrast with well-spaced falling dominoes though, "moments" are created by awareness, and need not happen with a regular cadence or frequency per se. Maybe some dominoes fall over immediately and others waver a little first...
Hopefully some of this helps?