# Which derivative of $s(t)$ can be discontinuous in an idealized world of physics?

I'm a ninth grader and I've yet to take my first physics course, also I've never taken a calculus course. In my physical science class, we were making qualitative graphs of distance over time, velocity over time, and acceleration over time using our intuition to describe how one function changing would affect the others. We also did some baby derivatives/integrals, deriving $$d=\frac {1}{2}at^2$$ and the like. Anyway the fact that many of the graphs had sharp turns got me thinking, is this really possible?

Obviously objects can't teleport so $$s(t)$$ must be continuous, and a sharp point on $$s(t)$$ would cause the graph of $$v(t)$$ to be discontinuous, meaning there is an undefined point on $$a(t)$$. We also learned that $$F=ma$$ so of course this would require an infinite amount of force, which seems impossible. So can $$a(t)$$ be discontinuous or is the jerk of an object the lowest derivative of $$s(t)$$ that can be discontinuous? Perhaps $$s(t)$$ functions must be able to have an infinite number of derivatives taken without eventually reaching a discontinuous function?

$$a(t)$$ can be discontinuous, in an idealized physics setup. Imagine sliding an object off a cliff—it drifts towards the edge with constant velocity and no net acceleration, but when it falls off the cliff, the acceleration suddenly changes to $$9.8$$ $$m/s^2$$. Of course, in the real world, such an object would have finite size and it would probably tip over the edge, and the acceleration would be continuous, albeit very rapidly increasing from $$0$$ to $$9.8$$.
If we were to make our setup even further idealized and abstract, we can even make $$v(t)$$ discontinuous! (In this case $$a(t)$$ ends up becoming something that's not even a function, but that is advanced material. Even though it's not really a function, we can do very useful math with it. Look up "Dirac delta function" if you're curious.) To achieve this, think about two very hard blocks colliding with each other and bouncing off. You could think of the $$v(t)$$ in this case as jumping from one value to another instantaneously.
Keep in mind that the above picture is abstract and mathematical, and doesn't fully capture all the physics. Of course, hard objects will deform slightly during a collision, and if you zoom in very closely on a very precisely measured $$v(t)$$ graph, it would be continuous but with "very large derivative" during the collision time. But the point of these mathematical abstractions isn't to paint a perfect physical picture, but to capture the most important physical ideas while keeping everything mathematically tractable.