Let, $\{q_n\}_{n \in \mathbb{N}}$ be an enumeration of rational numbers. Consider the function $f : \mathbb{R} \to \mathbb{R}$ given by, $$\displaystyle f(x) = \sum\limits_{n : q_n < x} c_n$$

where, $\displaystyle \sum\limits_{n=1}^{\infty} c_n$ is an absolutely convergent positive series. The function is clearly monotone increasing, discontinuous at rationals (with jump exactly $c_n$ at $x = q_n$) and continuous at irrationals.

$1.$ I wish to ask about the points of differentiability of $f$?

Since $f$ is monotone it should be differentiable a.e. but how do we identify these points of differentiability? (as in a way of representing this set in a compact way)

Intuitively, it seems they should be related to the particular enumeration of the rationals $\{q_n\}$ at hand. For example if we have an enumeration such that for $\alpha \in \mathbb{R \setminus Q}$, we have $q_n \notin (\alpha - \delta_N , \alpha + \delta_N)$ for $1 \le n \le N$ (i.e., say $|q_n - \alpha| > \delta_n$ for $n \in \mathbb{N}$ where, $\delta_n \downarrow 0^{+}$ as $n \to \infty$) and now if we impose further the 'nice' property:

$\displaystyle \frac{f(\alpha + \delta_N) - f(\alpha)}{\delta_N} = \frac{1}{\delta_N}\sum\limits_{n : q_n \in (\alpha, \alpha + \delta_N)} c_n \to \lambda$, (as $N \to \infty$) and similarly one for the left derivative, we have $f'(\alpha) = \lambda$.

So, intuitively I can see how to choose an enumeration that makes the derivative equal $\lambda$ at $x = \alpha$ (or blows up at $\alpha$, i.e., $\lambda = + \infty$).

To clarify what I am asking: Given an enumeration of rationals, how do we come up with relevant definitions/concepts relating to said enumeration, which helps us identify which $\alpha$'s we should expect to be a point of differentiablity.

$2.$ Is there a way to estimate the derivative at these points?

Has these questions been addressed/answered in literature before? I'd love it if I could get some reference in this matter. Thanks! :-)