Any physical system that makes/breaks contact, such as walking robots. Sure, the foot is not perfectly rigid and technically is a stiff spring. But from a computational perspective, problems still bear all the hallmarks of a discontinuous system such as requiring a very short integration step.

Yes, that's another example of the discontinuity existing in an idealized model, but not real life.

Quantized space is absolutely discontinuous, and tunneling is a discontinuous system. In fact assuming the universe is quantum it’s discontinuous in reality but the appearance is continuous. But these distinctions aren’t super useful unless you’re dealing with these sorts of effects. Continuity is the approximation, discontinuity is the reality. But depending on what’s useful we use the mathematics that help us.

Tunneling currents are continuous in every parameter, although I admit that when you're dealing with particles you have continuous probability distributions with continuously varying means, rather than continua of matter. (But that should count, because all macroscopic variables are expectation values.)

Specifically at quantized space and time levels everything is discrete even distribution functions. There’s no sense in having a continuous spacial distribution sub Planck lengths.

If I wanted to understand and obtain your intuition about linearity, what would you recommend?

I question how strongly I would recommend the years of working with it. :-)

Like pure math? Engineering?

Study it in school?

There are a lot of linear problems that I'm interested in.

Do you use a computer algebra system?

Do you have a few books?