Is a curve being continuous a necessary or sufficient condition for that curve to be a physical trajectory? In calculus classes, continuous functions are often intuitively described as functions whose graph can be drawn on a piece of paper without picking up your pencil.  I'm wondering how much that intuitive description has a basis in physics.
To make things simple, let's restrict things to classical mechanics and to point particles.  My question is, according to classical mechanics, is the trajectory of a point particle in three-dimensional space always a continuous curve?  And conversely, is every continuous curve in three-dimensional space a possible trajectory for a point particle?
I think the answer to the latter question may be no, given the existence of space-filling curves and the like.
 A: If there was some discontinuity in the trajectory of a particle, then the particle must experience an infinite velocity at some point in time. In reality this is impossible$^*$. Therefore, the trajectory of a particle must always be continuous. Mathematically, if the function is not continuous then it is not differentiable, and we run into issues with the velocity.
However, not all continuous curves are valid trajectories. The velocity must also be continuous. If there are any sharp corners or "kinks" in the curve, then an infinite acceleration would be required to travel on that path. Infinite accelerations are also not possible. Therefore, continuous does not mean possible trajectory. However, one caveat to this is that it depends on how this curve is traversed in time, as a curve/trajectory does not define a unique $\mathbf r(t)$. If the velocity becomes $0$ at the kink then we no longer require an infinite acceleration, and this becomes a valid, physical trajectory. An example of this would be a point on the edge of a rolling circle whose path is a cycloid that has a kink every time it touches the ground. Or an even simpler example is just an "L-shaped" trajectory where the particle comes in from one direction, stops, and then continues on in another direction.
Therefore, a continuous path is necessary but not sufficient for that path to be a valid, physical trajectory. However, a continuous path that is traversed such that the particle stops and starts at any kink is then sufficient to be a valid, physical trajectory.

There are certain examples where trajectories can go off to infinity in a finite amount of time. While these should be kept in mind, they seem to be present in specific scenarios, so I will not discuss them here.
