I have studied physics up to 12th grade and I noticed that whenever new equations are introduced for certain entities, such as a simple harmonic wave, we never prove that it's continuous everywhere or differentiable everywhere before using these properties.
For instance we commonly use this property that $v^2\cdot \frac{\partial^2f}{\partial x^2} = \frac{\partial^2f}{\partial t^2}$ holds for the equation to be a wave, and personally I've used this condition dozens of times to check if a function is a wave or not, but I've never been asked to check whether the function I'm analyzing itself is defined everywhere and has a defined double derivative everywhere.
Is there a reason for this? There are many more examples but this is the one I get off the top of my head.
Best Answer
Short answer: we don't know, but it works.
As the commented question points out, we still don't know if the world can be assumed to be smooth and differentiable everywhere. It may as well be discrete. We really don't have an answer for that (yet). And so what do physicist do, when they don't have a theoretical answer for something? They use Newton's flaming laser sword, a philosophical razor that says that "if it works, it's right enough". You can perform experiments on waves, harmonic oscillators, and the equation you wrote works. As one learns more physics, there are other equations, and for now we can perform experiments on pretty much all kind of things, and until you get really really weird as in black holes or smaller than electrons, the equations that we have give us the correct answer, therefore we keep using them.
Bonus question: let's suppose that, next year, we have a Theory of Everything that says that the universe is discrete and non-differentiable. Do you think the applicability of the wave equation would change? And what about the results, would they be less right?