It may or may not be helpful to realize that infinities (including infinitesimals) are merely a mathematical abstraction. Everything you encounter in the physical world is finite. Thus, it's not overly surprising that something actually happens, even though a given mathematical model of that something assigns it a zero probability.

That said, mathematical descriptions that include continuity are extremely convenient (life would be rather cumbersome if we had to use finite difference calculus instead of derivatives in all applications).

It is a very common tendency to identify a physical phenomenon with a particular mathematical model of it (one of the most abused models is that of virtual particles in particle physics), but one would be rather less wrong by keeping in mind that an abstraction of an object is not the object itself.

A nice (if fantastical) description of objects vs models can be found in the HPMoR chapter on partial transfiguration.