Why don't we prove that functions used in physics are continuous and differentiable?

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?

A lot of physicists would tell you that it doesn't matter if solutions to physical equations are smooth, as long as you can get meaningful predictions from them. Such a view is overly simplistic. There are circumstances where non-smooth features crop up in solutions to physical equations and are themselves very meaningful. The reason why high school physics classes don't worry about such matters is simply that they are typically beyond the scope of what can be taught in such a class.

A classic example of a meaningful discontinuity in a physical system is a shock wave. In certain (nonlinear) wave equations, you can have a solution that starts out smooth but eventually becomes discontinuous in finite time. These discontinuities tell you something useful: they can show up in real life as rogue waves in fluid dynamics or traffic jams in models of traffic. An example from Burgers' equation is shown below.

Solution to Burgers' equation, from Wikipedia

Discontinuities can form in many other systems, especially condensed matter systems, and indicate the presence of defects. Examples include vortices in superfluids (shown below) and dislocations in crystals. The ways that these defects behave often plays a dominant role in the overall behavior (i.e. thermodynamics) of the material.

One of the major reasons why it is useful to examine what happens when equations of physics break down is that these are precisely the circumstances where we can learn about new physics. For example, the behavior near discontinuities in nonlinear wave equations can be either diffusive (where the discontinuity gets smeared out in time) or dispersive (where the discontinuity radiates away as smaller waves), and knowing which it is tells you something about the microscopic structure of the fluid. For this reason, identifying where physical equations fail to be well-posed or self consistent is really important. There is a famous open problem in mathematics known as Navier-Stokes existence and smoothness, whose importance can be thought of in this way. If the Navier Stokes equations turn out to generate discontinuities in finite time, it could have profound implications for understanding turbulent phenomena.

One physical theory where mathematical rigor is especially far from established is quantum field theory. QFT famously has lots of calculations that spit out $\infty$ if done naively. The reasons for this are not fully understood, but we think it has something to do with the fact that there are more fundamental, as yet unknown theories that kick in at very small length scales. Another historical problem related to mathematical nonsense in QFT has to do with the Higgs boson: In absence of a Higgs boson, certain calculations in QFT give probabilities which are greater than 1, which is of course impossible. The energy scale at which these calculations started to break down not only told us that there was some physics we didn't understand yet--namely, there existed a new particle to be discovered--but also told us roughly what the particle's mass had to be.

So understanding the well-posedness of mathematical theories of physics is important. Why then don't people worry about this in high school physics? The answer is simply that our current theories of physics have been so well refined that our models for most everyday phenomena are totally consistent and produce no discontinuities. And the reason they never ask you to check that your solutions are sensible is just that they don't want you to get bored, because the answer is always yes.

In fact, there are some very general results in the mathematical fields of dynamical systems and partial differential equations which guarantee that most physics equations have unique, smooth solutions. Once you know some of these theorems, you don't even need to check that most solutions are smooth--you are guaranteed this by the structure of the equations themselves. (For example, the Picard-Lindelof theorem accomplishes this for most problems in Newtonian particle dynamics.)

Generally speaking, you can assume that the functions you deal with in high school physics are suitably well behaved. This is taken as given and most students will never question it, or even realise that there is anything to question - so well done to you for thinking about this issue.

Even in more advanced physics, there is a tendency not to worry about the finer points of mathematical models as long as they produce physically realistic outcomes that match experimental results. Most physicists will not question the fundamental assumptions of a model until and unless it predicts a singularity or a paradox or some other "pathological" outcome. And even then the short-term solution is often to avoid pathological results by restricting the domain in which the model is applied.

Mathematicians, by inclination and training, tend to be more careful. What the physicist sees as a focus on reality, the mathematician perceives as a lack of rigour. What is rigorous to the mathematician is overly fussy and pedantic to the physicist.

As an example, engineers and physicists will happily use the Dirac delta function, whereas a mathematician will point out that $\delta(x)$ is not actually a function (technically, it is a distribution) and treating it as if it were a function can lead to incorrect results. The mathematician says "if $\delta(x)$ is a function then what is the value of $\displaystyle \int_{-1}^{1} \delta(x)^2 dx$ ?". The physicist says "in what physical situation would I ever need to use such a bizarre integral ?".