What "kind of math" is used in the standard model? From the perspective of set theory, there is a large space of possible functions between different spaces. But in practically applicable models, we predominantly see a small set of operators used, e.g., addition, multiplication, negation, square roots.
Is it possible to give a high-level description of the "kind of math" that is used in the standard model? What operators are used? If we write the dependencies of quantities on each other as functions, are they all continuous? Differentiable? Are things like $\text{max}$ or $\text{argmax}$ used? Is there anything like a threshold function? Note that I'm not a physicist, so feel free to rephrase the question if the way I've asked it is too confused.
One thing that motivated this line of thought is the apparent existence of threshold-y looking functions in physics, like the behavior of water given its temperature. I intuitively expect that such behavior is emergent from underlying continuous laws, and that there's no true discontinuity in the standard model. So I'm trying to assess to what extent that intuition is justified.
 A: The "kind of math" a physical theory uses depends on the "kind of physicist" that's doing the theory.
There is a sliding scale of mathematical rigor in physics, where some will use little more than ordinary real analysis to do classical mechanics (cf. most introductory courses) and others will throw the full weight of differential and symplectic geometry at it (cf. e.g. the books by Arnold or Abraham and Marsden). Often there are even equivalent formulations that use different ways to describe the same theory. It is this way in most subfields, and quantum field theory and hence the Standard Model is no exception.
Also, you probably have a much too simple idea of how physical theories work.
It's not really meaningful to ask whether "the dependencies of quantities on each other" is always smooth or continuous or whatever - we can always start from a smooth function and define some new quantity that depends on it in a non-smooth way. And physical theories like the Standard Model are so complex in their many aspects that you can't really list all quantities that someone might be interested in or something like that. Additionally, we're doing quantum mechanics, so the "quantities" we might be interested in aren't just real numbers, there are operators on Hilbert spaces, functional differentations, etc., where notions like "smooth" are not that obvious (or even useful).
If you really want to understand the math that is used in quantum field theory, there isn't really any shortcut - you need to learn quantum field theory.
A: That's a hugely broad question, so I won't claim that my answer covers everything.

*

*group representation: everything is built with Noether's theorem, in terms of symmetries (of space-time, gauge...)

*Lie algebra: most symmetry groups are Lie groups

*Fock spaces: the space of possible vector states must take into account changing numbers of particles, so it usually a direct sum of Hilbert spaces to all possible powers

*unbouded operators and spectral theorem: studying eigenvalue problems in Fock space involves unbouded operators, which have a very complex spectrum

*distribution-valued operators: the operators mentioned above have elements that aren't simple numerical values

*distribution theory: see above

*asymptotic series: in perturbative QFT, the principle of least action is studied by an expansion of the action that is, or is assumed to be, an asymptotic series

*flow equations: during the renormalization process, the Lagrangian of the theory moves in a multi-dimensional space and can, along the way, meet attractive or repulsive points, and so on

*tensor algebra: this comes from the relativistic aspect of the theory (and is of course related to group representations above)

As for the operators in Fock Space, they have very complex behaviors that aren't completely understood from a strictly mathematical point of view. They contain distributions, so they have singularities and, due to the continuous infinite dimension nature of the space, things like trace or determinant are hard to define properly.
I'm sure that I'm forgetting things, this is only what I got to use when I worked in that field.
