Isn't a container of water actually a good example? Maybe swirling gently (laminar flow) is a better mental image than shaking vigorously (possibly discontinuous).
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.
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.
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".
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?
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 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.
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.
