Is any real-valued function in physics somehow continuous?

The canonical example of this is the apparent singularity that arises in spherical coordinates when you pass around the earth only to find that your longitude has gone from $0$ to $180$. Or for example the singularity that arises in the Laplacian with spherical coordinates. These are all non-physical and are a consequence of choosing a coordinate system.

A prime example of this kind of thing comes up in general relativity where you'll see singularities in your metric. In many situations these singularities are actually artifacts of the coordinate system chosen, see here for example: https://physics.stackexchange.com/questions/223549/coordinate-singularity-in-metric

here's a wonderful read on this subject: "What is a Singularity - Geroch" and the takeaway quote:

The presence or absence of a coordinate singularity is not a property of the spacetime itself, but rather of the physicist who has chosen the coordinates by which the spacetime is described.

However sometimes these singularities point at failings of a given theory, like ultraviolet catastrophes for example. See here. In particular if a singularity exists in all coordinate systems (i.e is diffeomorphism invariant), only then can we conclude that perhaps this is a failing of our current theory. The point is that nature should not care about our choice of coordinate system.

Discontinuous functions are fairly common.

What's the magnitude of the force between two point charges, or particles which can be considered point charges,

$$F=\frac{kq_1 q_2}{r^2}$$

Where $q$'s are the charges, $k$ is constant, and $r$ is the distance between them.

This is quite clearly discontinuous when the distance is is zero, and diverges as the particles get arbitrary close to each other. Same thing happens for Newtonian gravity.

In fact this causes a problem when taking a Fourier transform of the potential, which is usually 'fixed' by either physically giving the photon a mass and letting the mass go to zero afterward, or just admitting we can't know with arbitrary precision the exact form of the potential, our experiments aren't good enough, so they add an extra $e^{-ar}$ into the potential and let $a\to 0$ after the calculation.

The other thing that happens is when you want to thing about point sources, so physicists introduce the Dirac delta 'function', which I believe is more properly described as a distribution, can be obtained as the discontinuous limit of continuous functions.

This is a cleaned up version of some comments of mine on the original question.

Some bad behavior is removable, some is not. The behavior that is removable in some sense "already is", from the physicists' perspective. For example, sinc can be thought of as (a multiple of) the Fourier transform of the indicator function of some interval symmetric about zero. This is really only uniquely determined up to a.e. equivalence, so we may freely choose our "favorite" representative to be "the function".

To put it another way, we can identify the "physical version" of a function $f$ as $\frac{d}{dx} \int_a^x f(y) dy$. This is what you would get by averaging $f$ over smaller and smaller intervals containing $x$. It can happen that this limit doesn't exist. In this case $f$ "really does" have a singularity, and that must be addressed somehow for $f$ to have any physical meaning.

Because singularities almost never really exist in nature, usually the answer is that there is some gap between the model and the reality. For example, it could be that the "physical" equation has a "regularizing" term with a small coefficient that you are neglecting. In this case or similar cases, the "real" function might not have any actual discontinuity but there could be a scale separation at the position that your equation predicts a discontinuity. Understanding this scale separation, even if only through a somewhat unrealistic model, is useful.