We say that a function $f:\left[a,b\right] \to \mathbb{R}$ is a singular function or a devil's staircase if $f$ satisfies the following properties:
- $f$ is continuous;
- $f(a) < f(b)$;
- $f$ is non-decreasing on $\left[a,b\right]$;
- There exists a set $N \subset \left[a,b\right]$ (named exceptional set) of Lebesgue measure $0$ such that for all $x \in \left[a,b\right] \setminus N$ the derivative of $f$ in $x$ exists and is zero.
Possibly, the most know example is the devil's staircase constructed such that the ternary Cantor set is its exceptional set. (http://math.mit.edu/~katrin/teach/18.100/Devil%27s-Staircase.pdf). In this case, the exceptional set satisfies that $\dim_H(\mathcal{E}) = 0$. There are also examples where the exceptional set has $\dim_H(\mathcal{E}) = 1$ (M. Urbanski, On the Hausdorff dimension of invariant sets for expanding maps of a circle. Ergodic Theory Dynam. Systems 6(2):295-309, 1986.)
- Is it possible to construct a devil's staircase such that the exceptional set is countable?
- Is it possible to construct a devil's staircase such that the exceptional set is uncountable and $\dim_H(\mathcal{E}) = 0$?
- Are there any examples of similar functions to the devil's staircase such that the exceptional set is a Cantor set with positive Lebesgue measure?