If at some time $t$ there were a discontinuity in the velocity-time graph, then the acceleration would be infinite at $t$. So intuitively, it seems that the velocity-time graph must be continuous. I was wondering if all derivatives of the position-time graph are continuous functions (i.e. if the position-time graph is smooth) and if there was a way to prove it.
As you said, the next derivative of the velocity with respect to time is the acceleration. And the acceleration could in principle have a step somewhere due to a force starting to act on the object.
If you plot the time versus position of 'the high point on a see-saw', there is an abrupt change when end A goes from high to low (while end B goes from low to high). This is not entirely trickery, there are lots of useful items that exploit some kind of discontinuity (a toggle switch or an electronic astable 'flip-flop').
What happens when light reflects from a mirror? How can we deny that the path of the light is sharply kinked, i.e. 'not smooth' or that the velocity abruptly reverses, which (if a point particle were involved) would imply infinite acceleration?
As Newtonian mechanics applies to an object, so center-of-mass motion of the object is always smooth because Newton's laws apply; a 'reflection' of a ball occurs by distortion of the ball's shape over a short timespan, and that distortion generates force just like compressing a spring, and the force accelerates the ball. This ought not to be generalized, however. Some things are beyond the scope of Newtonian mechanics.