# History of the Darboux-Froda theorem

I am curious about the history of the so-called Darboux-Froda theorem, which is the following theorem:

a monotone function $$f:[0,1]\rightarrow \mathbb{R}$$ has at most countably many points of discontinuity.

This theorem can be found in Froda's thesis (see the references for 1) where Froda states that this theorem is implicit in Ch. VI of Darboux's 'Memoire on discontinuous functions' (which is correct, I would say); Froda also states that the Froda-Darboux theorem has been known for a long time (in 1929). I would like to know who first proved the Froda-Darboux theorem, perhaps with the conclusion formulated using a sequence (rather than countable sets).

## 1 Answer

Originally this was to be a comment, as I don’t really answer the question, but it’s gotten a bit long. At some later time I might look into this further and post the results here, at least if no one else sufficiently answers your question.

In the past I’ve tried to track down the origin of this result (tangentially, when working on other things) and strangely, I’ve found this result, despite being so well-known and found in pretty much any real analysis text, to have been ignored by historians. Incidentally, when I say “I’ve tried to track down”, I’m not talking about looking at a few Wikipedia pages and some superficial google searches, but literature searches of the depth found in answer 1 and answer 2 and answer 3.

In any event, I suspect pretty much any of Darboux, Dini, du Bois-Reymond, Cantor, Ascoli, Volterra, Weierstrass, Harnack, Peano, Scheeffer, and maybe a few others would have easily been able to formulate and prove this result in the mid 1870s to late 1880s if someone had brought the issue to their attention (i.e. can the discontinuity set of a monotone function be uncountable), and probably the majority of the people I’ve named knew the result by 1890. For instance, Scheeffer (Beispiel zu Theorem III [Example for Theorem III] on pp. 61−62 of this paper) showed in 1884 that any countable set can be the discontinuity set of a monotone function. Surely Baire, Borel, Lebesgue, and a few other French mathematicians knew the result by 1905 or so, and certainly Young knew it by 1907 or so (see this answer).

I don’t know how anyone who bothered to even minimally look into the history of this result could have conceived that it was originally due to Froda, and at such a late date, which the Wikipedia page you cite originally claimed. For instance, glance over Chapter 3 (pp. 184−229) of Hahn's 1921 book Theorie der Reellen Funktionen. Even if you can’t read German (and I can’t), it’s pretty clear that the huge assemblage of continuity and discontinuity minutia in this chapter leaves a result like “discontinuities of a monotone function form a countable set” in the dust. Moreover, neither Hahn nor this book were particularly unknown during the 1920s, unlike for example, Henry Blumberg, who published far-reaching extensions of Young’s 1907−1908 results that I cited above. Finally, see B. S. Thomson’s answer to Proof of Froda's theorem.