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.