This is a partial answer with the goal of clarifying the question.
The idea of continuity is not the same as the idea of a continuum. See Peter Smith’s answer to iblue’s question, "Why are the rational numbers not continuous".
As Smith mentions one can have a continuous function such as the identity function going from a domain containing only rationals to a co-domain also only containing rationals. However, the rationals are not a continuum because the set of rationals do not contain all their limit points. For example, we can construct a sequence of rational numbers approaching an irrational number that would not be in the set of rationals as closely as we please. Since the irrational number is not in the set of rationals, the rationals are not a continuum. They do not contain all their limit points or targets of sequences.
To see how this might work in the real world consider Zeno paradoxes. Although Zeno’s arrow gets closer to the target at each point of measurement, will the arrow actually reach the target? If the travelling arrow contained all its limit points then the target event should occur. Otherwise the moving arrow would be like the rational numbers lacking a limit point in the irrationals. For a mathematical presentation of this see Nathan Pflueger’s lecture “Convergence of series”.
Overmind’s answer to the question contains this idea of limit point. When the beam of light is measured it reaches the target, that is, the quantum system of the beam collapses. This collapse is certainly part of the real world since it is the only part of the beam we actually see.
Overmind makes the following philosophical assumption that is at the heart of the question:
I'd also say a basic particle is a continuous construct. Otherwise,
the particle would lose integrity.
The beam would be continuous, or rather a continuum, if that statement were true, but is it?
Unlike Zeno’s arrow which we can watch going to the target, we can’t see the beam go to the measuring device. Is the reality surrounding the measurement of the beam also a “basic particle”? If it isn’t then the target photon and whatever that beam was prior to acting like a photon at the target are not the same. If the beam does not contain its limit point as a set of reality like itself, then it would be like the rational numbers lacking an irrational limit point and the beam would not be continuous.
So to answer the question, Is anything truly continuous?, would require answering the question about what happens during a quantum collapse.