Does the position-time graph have to be a smooth function?

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.

• If it take a perfectly rigid ball (which is practically not possible), and bounce it off a perfectly hard surface, then it's velocity will be discontinuous in time. Are you asking whether such a situation is practically possible? – theindigamer Apr 5 '16 at 5:35
• I know that for velocity to be discontinuous there must be infinite rigidity, infinite force, or something else impractical. I am asking if all derivatives of position, i.e. acceleration, jerk, etc. also must be continuous. – Rogue Autodidact Apr 5 '16 at 6:11
• I think this is an interesting question. People are often very casual about how differentiable things need to be, and it is certainly useful to actually think about it carefully. My intuition is that everything is at least smooth but I don't know why I think that. One reason to ask for more than smooth is that, to do physics, you need to be able to approximate things in some nice way, by some power series say, and you need that series to converge. Well, if it's a power series then things need to be analytic. – tfb Apr 5 '16 at 6:45
• Related: physics.stackexchange.com/q/151399/2451 and links therein. – Qmechanic Apr 5 '16 at 6:48