If you look at the history of physics, including not just the final breakthroughs but the previous misunderstandings, there's a long history of "Well, we thought it had to be either A or B, but it turns out it's a novel and hard-to-imagine combination of both of those things."

For example, consider the universal speed limit that we now call the speed of light. Prior to relativity, when space and time were kept rigidly separated, there was a bit of a problem. The math of Newtonian physics and their corresponding frame transforms admit no particular speed limit; no matter how fast a thing is going, a force could always be applied to it to make it go yet faster. So a speed limit seemed to be impossible. But observationally, it became more and more clear there was one, and what's more, it seemed to be unrelated to your frame of reference. So apparently there was a speed limit, and not only that, but one that was essentially uncharacterizable in Newtonian physics. But if something was going the speed limit, what happened if you applied further forces on that thing?

The universe seemed contradictory; it seemed to have a speed limit, yet it couldn't have one, and if it did have one it seemed it would lead to contradictions as well. So, does the universe have a speed limit or not?

And the answer is basically that that question, in the context of a Newtonian view of the universe, is simply ill formed and has no meaning, because the universe isn't Newtonian. With the better (if not necessarily entirely correct) view that Einstein introduced, what we get is actually something that even now a lot of people don't really get, which is that the universe both does and does not have a speed limit. The former is generally understood; c is the limit. You can not witness anything in the universe going faster than c from your perspective. But there is also a sense in which the universe doesn't have a speed limit; no matter how fast you are going, you can always accelerate in any direction and go faster in that direction. There isn't a speed limit in the sense that there is no speed that you can be going where the universe will say "Nope, stop, you can't go any faster that way because physics won't allow it." (Other things, like blueshifting the cosmic microwave background into arbitrarily high-energy gamma rays from your point of view may stop you, but "physics" itself won't.) Subjectively, time dilation means that there is a sense in which you can head to distant location as fast as you like; you can cover what was originally a million light years in arbitrarily short amounts of subjective time, and there is a real sense in which that becomes your "speed" and it can greatly exceed lightspeed in your original reference frame. And just try explaining to a 17th century physicist the contortions the universe will go through in order to make it that even that arbitrarily-fast object from a certain point of view will still not witness anything going faster than light, with length contraction, time dilation, all that jazz. If you hadn't been taught it, you wouldn't ever come with it on your own.

That is, it's a complicated mixture of both having and not having a speed limit that, when fully understood, makes sense. But if you try to put the answer in the context of the original Newtonian understanding it makes no sense.

Personally I expect the base of our universe to have something similar when it comes to "is it continuous or discrete?" I expect it's going to end up being something that has elements of both, in some manner that is essentially impossible to understand if you insist on fitting it into that strict dichotomy but all works together and makes sense on its own terms. Loop quantum gravity is the best effort in this direction I've seen, but since it isn't "done" yet doesn't seem to be the right solution.

(There's been a lot of things in science history where there was some sort of "Is this thing rigidly This or That?" and the answer turns out to be "well, yes, and also no, it's more complicated and fluid than you thought" in some manner that would have been difficult to guess in advance.)