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! :-)

  • 2
    $\begingroup$ Asked yesterday on m.se, math.stackexchange.com/questions/1921071/… $\endgroup$ Sep 11 '16 at 22:32
  • $\begingroup$ @GerryMyerson thanks for the cross-link! $\endgroup$
    – r9m
    Sep 11 '16 at 22:33
  • $\begingroup$ I would think of it in terms of the $c_i$, except I would relabel the $c$'s and index them by the corresponding rational. Then you do the analysis independently of the enumeration, and then talk about how quickly or slowly a given enumeration "hits" an interval. Gerhard "Analysis Should Be Order Blind" Paseman, 2016.09.11. $\endgroup$ Sep 11 '16 at 22:38

Offhand, I do not know anything about determining in some concrete way the points of differentiability, but it is fairly well known that the points of differentiability form a meager set (i.e. a set of the first Baire category), and thus this situation gives us a natural example of a meager set that has full measure. It's possible that googling or using Math. Reviews for some of the references I give below will lead to papers more specific to what you're asking, but I haven't tried (and I do not have access to Math. Reviews).

W. H. Young [9] proved in 1903 that for any real-valued function defined on an interval, the set of points at which the function has at least one infinite Dini derivate is a $G_{\delta}$ set. Thus, if this set of points is dense in the interval, which will be the case for the function you're considering, then this set of points will be co-meager in the interval. In other words, at almost every (in the Baire category sense) point in the interval, the function will have at least one infinite Dini derivate. This of course implies that at almost every point (in the Baire category sense) in the interval, the function will not have a finite two-sided derivative, since the set of points where the function does not have a finite two-sided derivative is a superset of the set of points at which the function has at least one infinite Dini derivate.

Young's result was fairly well known during the next 30 years or so (in fact, Young's various results about $G_{\delta}$ sets is why people during this time often referred to $G_{\delta}$ sets as "Young's sets"), but it became less known as people began moving into other areas (such as abstract measure-theoretic notions and functional analysis) at the expense of classical real analysis and point set theory. Thus, as typically happens in these situations, the result was later rediscovered --- by Brudno [3] (Theorem I), Ćetković [4], Fort [6], Marcus [8] (Theorem 1) [all independently I believe, except that Marcus knew of Brudno's paper] --- and some of this later work was followed up by Boas/Cargo [2], Filipczak [5], Garg [7], and others. These references, except for Brudno [3], are given in a discussion on p. 157 of Boas [1]. As for [3], I discussed this paper briefly in my answer to the math Stackexchange question Characterization of sets of differentiability, at the end of the paragraph prefaced with "(ADDED NEXT DAY)".

[1] Ralph Philip Boas, A Primer of Real Functions, The Carus Mathematical Monographs #13, 4th edition revised and updated by Harold Philip Boas, Mathematical Association of America, 1996, xiv + 305 pages. MR 97f:26001; Zbl 865.26001

[2] Ralph Philip Boas and Gerald Thomas Cargo, Level sets of derivatives, Pacific Journal of Mathematics 83 #1 (July 1979), 37-44. MR 81d:26002; Zbl 424.26005

[3] Alexander L'vovich Brudno, Непрерывность и дифференцируемость [Continuity and differentiability], Matematiceskii Sbornik (N.S.) 13(55) #1 (1943), 119-134. MR 7,10a; Zbl 63.00636

[4] Simon [Simeon] Ćetković, Un théorème de la théorie des fonctions [A theorem in the theory of functions], Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences (Paris) 245 #20 (1957), 1692-1694. MR 19,946b; Zbl 80.27202

[5] Franciszek Mirosław Filipczak, On the derivative of a discontinuous function, Colloquium Mathematicum 13 #1 (1964), 73-79. MR 30 #3184; Zbl 129.04003

[6] Marion Kirk Fort, A theorem concerning functions discontinuous on a dense set, American Mathematical Monthly 58 #6 (June-July 1951), 408-410. MR1527895; Zbl 43.05503

[7] Krishna Murari Garg, On the derivability of functions discontinuous at a dense set, Académie de la République Populaire Roumaine. Revue de Mathématiques Pures et Appliquées [after 1963: Revue Roumaine de Mathématiques Pures et Appliquées] 7 #1 (1962), 175-179. MR 26 #2557; Zbl 117.28803

[8] Solomon Marcus, Points of discontinuity and points of differentiability (Russian), Académie de la République Populaire Roumaine. Revue de Mathématiques Pures et Appliquées [after 1963: Revue Roumaine de Mathématiques Pures et Appliquées] 2 (1957), 471-474. MR 20 #3946; Zbl 87.05002

[9] William Henry Young, On the infinite derivates of a function of a single real variable, Arkiv för Matematik, Astronomi och Fysik 1 (1903), 201-204. JFM 34.0411.02 [See here if the google-books link to the paper doesn't work for you.]


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