Forgive me for asking the obvious question, but isn't that… just "atoms"?
(And now I remember my excitement when I first got to play with satellite data and python, did a Fourier transform to see if there was any temporal pattern to the environment, and was disappointed to realise I'd just rediscovered "winter and summer" in quite possibly the most ridiculous way possible).
I guess what I’m trying to say is just that I’ve always been surprised by the lack of interest most pure mathematicians show towards questions of ‘what exists’ or ‘what the purpose of mathematics is’ and so on. I would have expected such (clearly) intellectually curious people to be less myopic.
Because the world is a big place, one can find countless essays on these sorts of topics. But it’s still a minority sport. John Baez is a brilliant outlier in many ways.
But successful applications of the real numbers are everywhere present in physics, engineering, statistics, and the other natural sciences. The funny thing is that many of those applications don't care either if real numbers represent an aspect of reality. They are a well-enough approximation, and that makes them work.
Of course, this is opening another whole can of worms that is perhaps starting to become off-topic.
You could write your problem as a discrete system and often that is kind of it. People have tried for a long time and the solution methods to the problems are almost impossible or too expensive.
Pure mathematics has indeed been challenged whether it actually makes sense to further fund it, but occasionally their objects of study turn out to be useful. For example number theory, which is the foundation of modern asymmetric cryptography.
But even if their research never turns out to be useful, pure mathematicians are unconcerned about it since they don't approach their research objects with any assumption regarding that.
While typically taught as chaotic, Newton's fractal is probably the most accessible example.
No matter how small your epsilon is, your piece of the continua will either contain one root or all roots.
It is an indecomposable continua.
Predator pray with fear and refuge is an example where you hit this in numerical systems.
In one of my papers, I actually assumed that two features in material have to be minimum epsilon distance away to show that the method convergences in a reasonable amount of computational effort. (where epsilon is arbitrarily small) So, I did break the continuum barrier to do some useful physics ones.