Is acceleration continuous? The extrapolation of this Phys.SE post.
It's obvious to me that velocity can't be discontinuous, as nothing can have infinite acceleration.
And it seems pretty likely that acceleration can't be discontinuous either - that jerk must also be finite.
All 4 fundamental forces are functions of distance so as the thing exerting the force approaches, the acceleration must gradually increase (even if that approach/increase is at an atomic, or sub-atomic level)
e.g. in a Newton's Cradle, the acceleration is still electro magnetic repulsion to it's a function of distance, so it's not changing instantaneously, however much we perceive the contact to be instantaneous. (Even if we ignored the non-rigidity of objects.)
Equally I suspect that a force can't truly "appear" at a fixed level. Suppose you switch on an electromagnet, if you take the scale down far enough, does the strength of the EM field "build up" from 0 to (not-0) continuously? or does it appear at the expected force?

Assuming I'm right, and acceleration is continuous, then jump straight to the infinite level of extrapolation ...
Is motion mathematically smooth?
Smooth: Smoothness: Being infinitely differentiable at all point.
 A: acceleration cannot be infinite.  Things need a force to be able to accelerate an object. So to infinitely accelerate and object it needs a infinite force. Also try wording question  better.
A: Not a physicist, but I think acceleration can be discontinuous. Consider a car travelling at constant velocity (acceleration = 0) that hits a wall. De-acceleration (negative acceleration) commences until the car comes to a complete stop. For all intents and purposes over time t acceleration starts at zero, decreases to a negative value (because de-acceleration), and then instantaneously jumps back to zero.
My 2 cents. 
A: With respect, I  think you're splitting hairs. 
Before the wall, the velocity is constant, and a = 0. After the wall, the velocity is constant at 0, and a = 0. In between velocity is decreasing to 0, acceleration is the derivative of the velocity as a function of time, at some non-constant value of un-changing sign (since it's either constantly decelerating, or accelerating), so how does it start at 0, end at 0, and have some value of unchanging sign in-between without being discontinuous? 
