The answer by @MauroGiliberti is great, but we do work with discontinuities in physics as the answer here says. In fact, a lot of careful and rigorous analysis is going on in general relativity, as there smoothness/singularity problems easily arise.
Newtonian physics however is very intuitive and easy. You do not have just some random mathematical entities, you have entities which are to describe real world.The math represents some mechanism and from intuition you know how the math should behave.
Take for example falling rock from height $h_0$. The equation of motion is $md^2h/dt^2=F,$ where F is the force. Do we need to show that $h$ is twice differentiable everywhere and that $F$ is function? Of course not, as we know how the system is supposed to behave. And it is not twice differentiable everywhere (and the force is not in fact function), since the movement of the rock is described by this function:
where $H$ is heaviside step function.
From mechanism of gravitation we know that before the rock hits the ground, the system is supposed to be well behaved and we also know what happens when the rock hits the ground. Because of this, you never see analysis like this in a physics class, where you would use discontinuous heaviside step function in solution to simple fall of the rock.
I've never been asked to check whether the function I'm analyzing itself is defined everywhere
Why would it need to be defined everywhere? When you analyse wave, you care about the thing you observe. You do not care what is going on with this wave in the other side of the universe. The computation thus better be independent on what goes on in there.
The physicist just have some idea about the mechanism how the universe is supposed to work, and have some intuitive understanding why the math he is using is supposed to correctly represent it. Then he can just assume the functions are well behaved, as physics demands. Sometimes he even uses the math knowingly incorrectly, because he might have reasons to think this incorrect manipulation does represent the mechanism he has in mind.
Then he just checks wheter the results agree with experiments. If they do, he will create work for many many mathematicians trying to make some sense of what he did. And they are not always successful. Take for example statistical physics. It is 100 years old, produced enormous amount of evidence that it works, yet mathematicians are still struggling to show the calculations are in fact consequence of the known laws of physics.