Extension of the Derivative Due to the definition of the derivative, non-continuous points aren't differentiable. Interpreted as the slope, it wouldn't make much sense most of the time, either. But I was wondering if one could extend the definition of a derivative to act as it does, but also be defined for non-continuous points (similar to the gamma function as an extension for factorial). While most of the time it might be useless or more complicated than necessary, I thought it might give rise to some interesting trains of thought investigating the possibility. So is there an extension of it, or any thought into the matter?
 A: I don't know of any work relating to your question, so I will just throw out some thoughts. :) People better versed in analysis could perhaps write something more interesting, rigorous, and principled!
First of all, note that continuity does not imply differentiability, but the converse implication is true.
For instance $|x|$ is not differentiable at 0 and the Weierstrass function is differentiable nowhere, but both are continuous everywhere. (In fact, most continuous functions are nowhere differentiable). 
So already continuity is too weak to allow defining a derivative. 
However, one could invoke the Weierstrass Approximation theorem, which says that any continuous function can be arbitrarily closely approximated by a polynomial. Obviously, a derivative can be defined for a polynomial, so if $f$ is a continuous function which we approximate $\epsilon$ closely with a polynomial $\tilde{f}_\epsilon$, we could (non-rigorously) "say" its "extended $\epsilon$ derivative" (or some such name) is $\partial_{x,\epsilon} f(x):=\partial_x \tilde{f}_\epsilon(x)$. Using Lusin's theorem, we can extend to measurable functions (choosing a continuous function it coincides with everywhere outside a set of arbitrarily small measure). Similarly, the continuous functions are dense in $L_p$ so we could do something similar for those functions. The list goes on. (At some point we will run into problems though! e.g. see here).
Now let's think about extending to discontinuities. More directly, suppose we have a function that is piecewise continuous with a countable number of jump discontinuities (at $r_i$, $i\in\mathbb{Z}$). As the commenter noted above, it's not obvious how to give a derivative at $r_i$. There are many ways one could imagine to do this, but the most natural to me is to use splines to smooth over
these points. However, note that the error in the function approximation can now be arbitrarily large! One could also use the average of the approximate derivatives on each side (but then it won't be continuously differentiable, in our approximate sense).
There are no doubt many other ways to imagine such an extension (maybe using Fourier series to smooth discontinuities instead). However, I will note that approximating derivatives in noisy, discontinuous data is a common applied math problem  (e.g. estimating manifold curvatures from noisy data, for instance). Often it is actually solved by fitting local multivariate approximations and taking their derivatives. So this "extension" is not totally unreasonable, however non-rigorous and unsatisfying to the real analyst. 
