In the new issue of Analysis that Brian just mentioned, there’s an article by Hud Hudson called “How to Part Ways Smoothly”. He describes two point-objects that are colocated at every time before 100, but then are at a different location at that time, although both move continuously. The way this works is that both rotate around a clock face doubling their speed after each rotation, so that they go around infinitely many times before 100, and that at 100 one of them is at the 3 on the clock face and the other is at the 9.
I don’t understand why he is justified to claim that “neither character ever moves discontinuously”. It’s true as he says that no matter how small an interval you look at before time 100, there are time-slices of each character that are arbitrarily close to their destination. However, I normally think of continuity as being characterized by a different set of quantifiers – for every spatial distance, there is some temporal duration such that all time-slices within that duration are within that distance of the destination.
Because their trajectories satisfy Hudson’s definition of continuity but not mine, the space-time trajectories are said to be “connected, but not path-connected”, and curves like this are standard counterexamples in topology. But is there any reason why metaphysicians might adopt Hudson’s account of continuous motion and not mine? If not, then an example like his could be constructed whereby a particle traces out successive approximations to a Peano space-filling curve with constantly doubling speed, so that in the limit every point in space could be the result of “continuous” motion.