If a function is continuous but nonsmooth at point x = a, but has a finite derivative = 4 at x = a - $\epsilon$ and a finite derivative = 7 at $x = a + \epsilon$, both in the limit $\epsilon \to 0$.
Under what conditions (if any) is it appropriate to write that the derivative at x=a is $3 * \delta(x - a)$?
Or do we say the derivative is infinity at $x = a$, or do we say the derivative is undefined at $x = a$?
We can say (in the distributional sense) that the derivative of a "function" is $3\delta(x-a)$ if there's a jump discontinuity of height $3$ at $x=a$.
In your case I would say it's more appropriate to simply say that the derivative is undefined there.