Consider the probability distribution of a real-valued R.V. as the equivalence class of generalized PDFs where the integral over each measurable set in $\mathbb{R}$ is the same in each PDF.
1) Can any R.V.'s distribution be represented as the sum of a normal function and a countable number of $\delta$'s?
2) If so, does there exist an element in the equivalence class where the 'normal function' part's set of discontinuities is nowhere dense?
I am trying to work out the form of a general product distribution and it would be helpful to know what distributions look like.
There exist distributions which are neither discrete nor continuous. For example, the Lebesgue-Stieltjes measure generated by the Cantor function.