The concept of differentiability is fundamental to Physics. For instance, already second Newton's law $$\mathbf{F} = m \frac{\mathrm{d}^2 s}{\mathrm{d}t^2}$$ involves the second derivative of space respect to time, so it is implicitly assuming that space as a function of time is differentiable at least two times.
I wonder: Has the question of how many times the fundamental physical quantities are differentiable (respect to time or other fundamental quantities) been investigated?
I understand that:
The answer may depend on the branch of Physics (classical, relativistic, quantum...).
It is important not to conflate the physical model with the actual phenomenon. For instance, already in classical mechanics we have "non-differentiable" velocities when it comes to perfect elastic bounces, but this seems just a limit of the model. Indeed, better models (dropping the "perfect" assumption) restore differentiability.
High order derivatives of position are used in engineering (up to the 6th derivative of position), but this is done in a pragmatic way, the question if, say, position is really derivable 100 times is not considered.
