I think there's an easier thought experiment to generate non-determinism in classical mechanics without some continuity assumption:
Take a chaotic system (eg., the moon of one of our solar system planets) and let it evolve for some time, T. Track the position with coordinate X. Let T be large enough that the nth decimal place of X_T is significant to determining X_T+1.
If there is a discontinuity at the nth decimal place, then X_T+1 is not determined by X_T.
For quite observable T, n quickly becomes "sub-quantum". So, if classical mechanics is deterministic, and describes nature, nature must be continuous at arbitary depth.
> So, if classical mechanics is deterministic, and describes nature, nature must be continuous at arbitary depth.
Isn't that the classical assumption though? That nature is analogue. To avoid problems with infinities, you can just say let's assume it's continuous up to Graham's number or something.
Or, .. it's continuous but "unknowable" in the sense of becoming increasingly "out of focus" wrt measurement due to the uncertainty principle at and below the order of Planck's constant.
Not every dynamic system is chaotic, and not system with chaotic elements is chaotic everywhere ... but ...
Even quite simple systems can have chaotic regions .. and within those regions two 'particles' (or phase space initial conditions) can startout arbitrarily close (within a fuzzy out of focus impossible to measure Planck distance) and end up nowhere near each other .. (ie not continuous).
This is a mathematical result of dynamic systems and can be arrived either by Lorentz's reasoning ( "the butterfly effect" ) or via Smale's Horshoe Map (taffy folding to infinity!).
Take a chaotic system (eg., the moon of one of our solar system planets) and let it evolve for some time, T. Track the position with coordinate X. Let T be large enough that the nth decimal place of X_T is significant to determining X_T+1.
If there is a discontinuity at the nth decimal place, then X_T+1 is not determined by X_T.
For quite observable T, n quickly becomes "sub-quantum". So, if classical mechanics is deterministic, and describes nature, nature must be continuous at arbitary depth.
OR: *classical* mechanics is non-deterministic.