Suppose we have a cumulative distribution function $F(x)$
Then by Lebesgue decomposition theorem, we may decompose $F$ into: absolutely continuous part, singular part, piecewise step part.
As for the absolutely continuous function, traditional notion of derivative works.
As for the piecewise step part, we can make probability density "function" using the notion of Dirac delta "function".
Then what is left over is the singular continuous part.
I am wondering if there is a way to extend the notion of a function in order to take the derivative of the singular part so that it makes sense under an integral so that the fundamental theorem is saved?
For example, what would be the "derivative" of the Cantor function so that the integral of the derivative would recover the original function?