Last week I worked through some kinematics, and it led to a discussion in lecture about the jerk, that is, the time derivative of the acceleration. I was emphasizing that the velocity must be a continuous function of time (lest we have infinite accelerations and hence infinite forces), but then I realized that most people in the class (physics majors) felt that the acceleration must also be a continuous function of time.
After a bit of thought I agreed with them, but not for the reasons they wanted; indeed several were unsatisfied with my discussion: As far as I can tell, the only thing that limits the jerk is the propagation of information. When you slam on the brakes on your car, it takes a finite time for the brake shoe to come in contact with the wheel. This is not really a fundamental problem with infinite jerk (the jerk appears in no physical law), but in any real situation because information propagation (and other kinds of changes) happen at finite speeds, the jerk can never really be infinite.