# Who introduced the terminology “nondecreasing” for weakly increasing (i.e. x≤y ⇒ f(x)≤f(y)), and when/why?

Arguably one of the most hated parts of English mathematical terminology is the word “nondecreasing”, referring to a function such that $$x\leq y \;\Rightarrow\; f(x)\leq f(y)$$ (what other conventions or languages might call something like “weakly increasing”, “(weak-)order-preserving” or various variations thereof). The term is much hated, of course, because it is not the negation of “decreasing” (i.e., there are functions which are not decreasing but neither are they nondecreasing), so the question arises as to how this annoyance came about.

I can form two (noncontradictory!) conjectures as to the origin of “nondecreasing”:

• One is that it is simply formed by analogy with “nonnegative” to mean “$$\geq 0$$” (which is also questionable, but at least makes sense logically) and “no less” to mean “$$\geq$$”: simply following the same pattern of using “non-” with the reversed inequality gives “nondecreasing”.

• Another is that “nondecreasing” should really be “monotonically nondecreasing” (some people indeed use the full form, which is cumbersome but clear), and that the adverb was eventually left out as it seemed redundant (rather than being later added by people who wanted to make the term less illogical).

Question: Who introduced “nondecreasing”, when, and why? And is either (or both) of the above conjectures correct?

• I've also seen "increasing" for $\le$ and "strictly increasing" for $<$. Googling shows this distinction is not uncommon (another much-hated nonnegative?). Sep 29, 2023 at 10:48
• Since "nondecreasing" has been used at least since the 1800s, I suspect this is not something that can be reasonably determined, since just because someone used it in 1866 (for example) doesn't mean that anyone else's later use was derived from that 1866 usage, and this is an especially relevant issue for something this long ago, when "transmission of knowledge" was much less than in later years with improved travel (e.g. trains, cars, planes) and improved communication (motorized mail travel, telephone, etc.), and not remotely like it has been in the past 30 years or so with the internet. Sep 29, 2023 at 11:01
• Regarding "has been used at least since the 1800s", see this google-books search, which I forgot to include in my earlier comment, and now can't include because of the $600$ character limit for a comment (and I don't want to omit anything I said in that comment). Sep 29, 2023 at 11:03
• @DaveLRenfro Some of the google books hits are for 20th century hits to journal articles to which g-b gives the date of founding of the journal (at volume 1), not the date of publication. Sep 29, 2023 at 11:19
• @kimchi lover: Golly shucks jeepers! OK, here's a definite early appearance of "nondecreasing" -- Note on the convergence of a sequence of functions of a certain type by Buchanan/Hildebrandt (1908; see also here) uses "nondecreasing" in (at least) 7 places and "nonincreasing" at least once (footnote, first page). This is the earliest use I can find from quick searches using google-books and JSTOR. Sep 29, 2023 at 16:28

My guess is that "nondecreasing" became a thing in the late 1800's and early 1900's, with the development of the theory of functions of bounded variation and of the Stieltjes integral. Thus, in his 1882 "Sur la série de Fourier" Camille Jordan wrote

Le théorème subsistera donc tous le fois que $$F(x)$$ pourra être représenté de $$0$$ à $$\epsilon$$ par $$f(x)-\phi(x)$$, $$f(x)$$ et $$\phi(x)$$ étant deux fonctions fines et non décroissantes.

Hobson's 1907 Theory of Functions of a Real Variable says the same thing (on p.336 of the 1956 Dover reprint):

... that it can be expressed as the difference of two bounded monotone functions, which are either both non-increasing or both non-diminishing.

According to F. Riesz and B. Sz-Nagy, writing in 1955 (on p.106 of the 1995 English translation of Functional Analysis), this result was made popular by Riesz's 1909 result about the dual of $$C[0,1]$$, for which it is useful to know (on p.10)

Every function of bounded variation is the difference of two nondecreasing functions.

My guess is that the English term "nondecreasing" is a direct borrowing from the French "non décroissantes". I do not know if the ambiguities the OP finds in the English term are present in the French one.

• Regarding Hobson's book, the 1st edition was published in 1907. The book was divided into 2 volumes for later editions. The 2nd edition of Volume I was published in 1921 and the 2nd edition of Volume II was published in 1926. The 3rd edition of Volume I was published in 1927 and no later edition of either volume was published. The Dover edition of Volume I (1927 edition) and Volume II (1926 edition) were both published in 1957. The excerpt you gave is on p. 316 of the 1921 edition and p. 336 of the 1927/1958(Vol. I) editions. The original version of the excerpt (continued) Sep 30, 2023 at 12:47
• is on p. 256 of the 1907 edition, but only the term "monotone" is used, modified afterwards by "neither of which diminishes as the variable increases". FYI, both volumes of my copy of the Dover edition (each priced $3.00) are dated 1957 (you used 1956). Incidentally, (+1) because I suspect if Jordan isn't the earliest, then he's surely a strong candidate for the term spreading to others given the influence of Jordan's Cours d'Analyse texts. Sep 30, 2023 at 12:54 • @DaveLRenfro Thanks for catching the 1956/1957 glitch, and being a bit more precise about Hobson's precise wording. Sep 30, 2023 at 13:14 • J. J. Sylvester, "Sur les actions mutuelles des formes invariantives dérivées." Journal für die reine und angewandte Mathematik, Vol. 85, No. 2, 1878, pp. 89 - 114. On p. 103 (my bolding): "cette fonction ordonné suivant les puissances ascendantes de$t$présentera une série de coefficients numériques distribués symétriquement autour de son milieu et ayant des valeurs non décroissantes depuis l une des extrémités jusqu au terme unique ou jusqu aux deux termes qui forment le milieu de la série " Oct 4, 2023 at 0:47 • Frédéric Frenet, Recueil d'exercises sur le calcul infinitesimal 2nd ed. Paris: Gauthier-Villars 1866, p. 2: "1° La somme de cette série, sachant qu'elle est indépendante des quantités positives non décroissantes$h_1, h_2, \ldots h_n, \ldots\$;" The earliest instances ("nichtabnehmend") in German publications that I could find date to the 20th century. So I think it is safe to say the term originated in French mathematics. Oct 4, 2023 at 0:58