How to use "$f$ is increasing when $f'(x)>0$" when $f$ involves the characteristic function.

Question

I am reading a solution to Folland exercise 3.30 which is "construct an increasing function on $\mathbb{R}$ whose set of discontinuities is $\mathbb{Q}$. In the solution, we enumerate the rationals and define $f=\sum_{n=1}^\infty2^{-n}\chi_{[q_n,\infty)}$, and $f_n$ to be the "stuff" inside the sum. Certainly, $f$ is an increasing function.
But, I was wondering how to take the derivative of the characteristic function, and if we can, could we just apply Calc 1 to say that if $f'(x)>0$, then $f$ is increasing? I know Folland has the Fundamental Theorem of Calculus for Lebesgue Integrals on age $106$ of his book, but I wasn't sure if we could do something like this too. That is, extend something simple from basic Calculus. Maybe this is something simple, and I am just not thinking of it...

Answer 1

Hard to figure out (for me) what you want to know. Certainly the derivative of your function $f$ fails to exist at each of the jumps. So you do not have a positive derivative, unless you are imagining that this function has either a positive finite derivative or a positive infinite derivative everywhere. (It does not.) I doubt anything from basic calculus prepares you to think about derivatives of saltus functions.

But, considering your topic is about "derivatives" and "saltus functions" I would guess this paper would be of interest:

Kuba, Gerald. On the differentiability of certain saltus functions. Colloq. Math. 125 (2011), no. 1, 15–30.

Exerpt from the Math Review: "As is well known, every monotonic function has at most countably many discontinuities and is differentiable almost everywhere. The standard example of a monotonic function which is discontinuous precisely at the rationals is given by $$F_φ(x)=∑_{φ(n)<x} \frac{1}{2^n},$$ where $φ$ is a bijection from $N$ onto $Q$. In this paper, the author investigates the differentiability properties of the functions $F_φ$. Some of his results are:

In every nontrivial interval there exists a nowhere dense null set $Z$ of irrational numbers such that every $F_φ$ has uncountably many points of nondifferentiability in $Z$.

If a one-sided derivative of an $F_φ$ exists at a point, then it is either infinite or it vanishes.

For every countable set $C$ there exists an $F_φ$ whose derivative exists (and vanishes) at every point of the set $C$. ..."