Shaking a container is a good example I think, assuming the glass is convex.

Erratic movement only occurs if the container isn't filled completely. If it is filled completely and the shaking doesn't include any turning/twisting motion, not much happens.

Even then. The required mapping is 'onto' (otherwise, a discontinuous counterexample would be trivial). The air particles are equivalent to the water.

The paths are continuous, but if they move two neighbouring molecules away from each other, the final transformation won't be continuous, will it?

It’s difficult to discuss this physical example because particles are discrete.

In an ideal system with points instead of particles, shaking would be continuous.

And then, we would not call it shaking but bending and twisting, would we?

Think of the 1D variant. If you shuffle a deck of cards, but require that to be ‘continuous’, few shuffles remain (I think only the identity mapping and ‘flipping the deck upside down’). I doubt anybody would restricting the possible permutations that much stil call shuffling.

I don’t think that analogy works because an idealized fluid is point particles, but an idealized deck of cards isn’t.

They both have a concept of neighbors but the deck of cards is simpler, it’s a good illustration, no?

I think point particles are different because there are infinitely many of them. They are more continuous than cards. This means shaking isn’t necessarily discontinuous even if two particles that were “right next to each other” wind up far apart, there should be more particles in between that ended up closer.

It doesn't make sense to me because a glass of water is a discrete set. If we only had two molecules you could just interchange them (with a 180 degree rotation), without any fixed points.

I'd guess the example of the glass of water was made as an afterthought, or as a vivid example of the potential complexity of the theorem. As your example shows, it simply cannot be true. If you think of a vase with marbles, it is also clear. Or if you think of an (ideal) gas in a closed system: would there be at least one particle that always(!) stays in the same place?

Show me a mathematical continuum in the real-world. And this is just the domain, nothing said about the mapping yet (the continuous function). Every analogy has its limits.

The space represented in a flat piece of paper.

The space is continuous even if we can only measure down to the Planck length.

Key here: "represented"

I'm only a mathematician, no physicist. But I think to remember, that the concept of a continuous physical space becomes quite muddled at this scale.

* * *

If you get to assume continuous paper, then I get assumed continuous paper.

If you don't want to assume continuity, then you get it back by rephrasing your theorems as "within a margin of error equal to the distance between discrete objects".