Conditions for this functional relating densities under change of variables to exist? Suppose I have a random variable $X$ with density function $f_X(x)$, and a continuous but non-smooth function $g$. We will also take $Y := g(X)$ to have a smooth density function $f_Y(y)$.
If $g$ had been smooth then we could have considered the equality
$$f_{Y}(y) = \sum_{k=1}^{n(y)} \left| \frac{d}{dy} g^{-1}_k (y) \right| f_{X}(g^{-1}_k(y))$$
where $n(y)$ is the number of solutions in $x$ for $g(x)=y$.
But since $\frac{d}{dy} g^{-1}_k$ is undefined, I wonder if we can suitably generalize to a functional $h$
$$f_{Y}(y) = \sum_{k=1}^{n(y)} \left| h \circ g^{-1}_k (y) \right| f_{X}(g^{-1}_k(y))$$
that plays a similar role to the derivative. I guess a weak derivative might qualify, but I am wondering if there are valid choices where $h$ is not a weak derivative. An especially desirable, but difficult, case to deal with is when the number of non-smooth points is uncountable.
Under what conditions can we say that $h$ will exist?
 A: I find it difficult to imagine what sort of practical application this has.
Yes, in pure mathematics among the set of smooth functions the complement of the set of functions that are non-differentiable almost everywhere are a meagre set (Sur les fonctions non dérivables Mazurkiewicz); ie differentiable functions are the exception.
But in practice we we always deal with functions that are differentiable. Either reality is differentiable, or for practical purposes it is enough to use differentiable functions as approximations of non-differentiable functions.
Maybe, what we consider as 'density' should not be considered as a function that expresses a property in a single point, and it is more a property of small region. (e.g. the density of some material is incorrect when you observe the material at an atomic level, it wil have peaks of high concentration and in between large amounts of empty space)
If you have a non-differentiable function, then possibly you may use some numerical differentiation function as substitute for the derivative. That is going back to the concept of a derivative that we learned as students. The simplest example is the function $$f^\prime(x) \approx \frac{f\left(x+ \frac{1}{2} h \right) - f\left(x- \frac{1}{2} h \right)}{h}$$
This is what the derivative means to us in a practical sense, and I guess it won't be bad to use this if you only care about the practical applied mathematics.
