I am working on tomographic methods in which the data is the "distribution" of values along a line rather than an integral. Given a measurable function $f:[0,1] \rightarrow \mathbb{R}$ one can define the push forward measure $\mu(s)=\lambda(f^{-1}S)$ for a measurable set $S \subset \mathbb{R}$ where $\lambda$ is the Lebesgue measure on $[0,1]$.
The density function $\phi$ is the Radon-Nikodym derivative $d\mu/d\lambda$, when that exists, but more generally it is the distributional derivative of the cumulative distribution function $\mu(-\infty,y]$ with respect to $y$, or simply the generalized function associated with the measure $\mu$.
For example if $f$ is constant $c$ then $\phi(y) = \delta(y-c)$ and if $y$ is a non-critcal value of $f$ and $f^{-1}(y)$ is finite $$ \phi(y) = \sum_\limits{x \in f^{-1}(y) } \frac{1}{f'(x)}.$$
In the case of an isolated critical point $x$ with $f^{(k)}(x)=0$, $0<k<m$ and $f(x+h) =y + f^{(m)}(x)h^m/m! +o(h^m)$ one gets a singularity in $\phi$ at $y$.
Is there a complete description of $\phi$ for any smooth functions $f$ in the literature? What about $[0,1]$ replaced by the closure of a domain with a smooth boundary in $\mathbb{R}^n$.
One difficulty is that the terminology makes it difficult to search for. Distribution also means generalized function, and in the probability literature where one can search for "probability density function" they are not that interested in random variables that are a smooths functions.