Define $c(x)$ is the Cantor function, which is monotone increasing and continuous(with no jump) in $[0,1]$.
We know $c'(x)$ is $0$ almost everywhere. And $c'(x)=\infty$ for $x$ belong to Cantor set. I just want to know how singular of $c(x)$ compared to a unit jump, i.e. Heaviside step function $\Theta(x)$. That is to ask what's $c'(x)$ as a distribution.
For example, we know $\Theta'(x)$ as a distribution is Dirac delta function $\delta(x)$: $$\int_{-a}^{a}f(x)d\Theta(x)=\int_{-a}^{a}f(x) \delta(x) dx =f(0)$$ with $a>0$.
So to see how singular is equivalent to find Riemann–Stieltjes integral of following function.
What's the Riemann–Stieltjes integral of
$$\int_0^x f(x')dc(x')$$ with $0\le x \le 1$. Especially, $x=1$.
Obviously $f(x)=1$, $\int_0^{x}dc(x')=c(x)$ and $\int_0^{1}dc(x')=c(1)=1$.
The distributional derivative of the Cantor staircase function is the Cantor measure, which is the measure that assigns value $1/2^n$ to each of the $2^n$ intervals of the $n$th generation of the set. The Wikipedia article is written from the probability point of which, from which the Cantor staircase is the c.d.f. of the Cantor measure.
The Cantor measure is the weak* (also known as "vague") limit of absolutely continuous measures $(3/2)^n \chi_{C_n}(x)\,dx$, where $C_n$ is the $n$th generation of the Cantor set, and $\chi$ is the characteristic function $(0-1)$ of this generation. That is, restrict the Lebesgue measure to $C_n$ and multiply it by $(3/2)^n$ to make the total mass $1$. Then take the vague limit as $n\to\infty$.
One can also view the Cantor measure as a product measure.
To integrate a continuous function with respect to the Cantor measure, one can follow the limiting process from the second paragraph: integrate over $C_n$, multiply by $(3/2)^n$, pass to the limit. How feasible the computation will be depends on $f$. In the special case of function $f(x)=(x-1/2)^n$ we get the central moments of Cantor measure, which are known and are expressed in terms of Bernoulli numbers. For another example, see Calculation of the $s$-energy of the Middle Third Cantor Set.