Why might singular functions be important in the study of the Fractional Quantum Hall Effect? In studying a course on Quantum Mechanics, I was led to this page, which explains that a singular function is one which is only non-differentiable on a measure zero subset of $\mathbb{R}$, at all points where it is differentiable it has derivative zero and yet it is non-constant! This is a very interesting property from a mathematical perspective but unlikely to have a physical meaning (or so I thought!).
However, on reading the Wikipedia page it says that indeed singular functions in the sense above do have relevance in Quantum Mechanics and in some way relate to the fractional Quantum Hall effect. I am actually studying for a Mathematics degree and so find the idea of Measure Theory appearing in even a minor sense in serious physics very intriguing! Anything similar to the Cantor function having physical relevance sounds fascinating...
Unfortunately, the page gives no references for this claim and I could not find any references online discussing where singular functions come up.
Could somebody give an example of such singular functions coming up in the study of the fractional quantum hall effect as mentioned above? Preferably any answer will come with a basic explanation of the investigations that led to these functions arising and any physical significance to the functions being singular.
 A: This is really just a comment, but I am posting it as an answer since comments don't display pictures.
When I saw the circle map plot in your first Wikipedia link, and given that you are asking about the FQHE, my mind immediately went to this very well-known plot of the Hall resistance showing various FQH plateaus:

While this does have the characteristic "staircase" shape, it's obviously not a singular function in the sense you're asking about, so I dismissed this idea pretty quickly.
However, after going back and looking at the plot of the Cantor function, it again reminded me very much of the staircase plot of $R_H$. So much so that I started to be convinced that this is the sense in which the Wikipedia author suggests singular functions appear in the FQHE.
As I said, the Hall resistance cannot literally be a singular function. For this we would need (at least!) for there to be a FQHE state for every rational filling fraction, which is not the case. But I wonder if the Wikipedia author meant to give a non-example which is merely evocative of other examples of singular functions, rather than providing an actual example.
