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I have two related questions:

  1. Who first defined the oscillation function (perhaps under a different name)?
  2. When did the switch from the phrase "saltus function"(*) to "oscillation function" happened?

Recall that given a topological space $X$, a metric space $M$, and a function $f:X\to M$, the oscillation function of $f$ is defined as:

$$\mathrm{osc}_\cdot(f):X\rightarrow[0,\infty],x\mapsto\mathrm{inf}\{\mathrm{diam}(f(U))|U\text{ is a nbdgd of } x\}$$


For what it's worth, Froda, in his 1929 thesis Sur la distribution des propriétés de voisinage des fonctions de variables réelles (https://eudml.org/doc/192780) attributes (p.23) the observation that the higher order oscillations stabilize to a 1905 paper by Denjoy (where this is used implicitly, Froda says as far as I can see) and a 1910 paper by Sierpinski.

(There are other, non-pointwise definitions, or else definitions in terms of the difference between limit superiors and limit inferiors also; these definitions are also admissible for the purposes of this question.)


(*): See e.g. Hobson's 1921 book The theory of Functions of a Real Variable and the Theory of Fourier's Series, Vol. I (2e) , p.291, or Blumberg's 1917 paper "Certain General Properties of Functions".

Finally, here is the OED definition of "saltus" (I hadn't heard of this word before):

enter image description here

(Oxford English Dictionary, s.v. “saltus (n.),” July 2023, https://doi.org/10.1093/OED/6909159924. )

In light of the definition of the word it seems more appropriate to call the function at hand the saltus function than the oscillation function (also keeping in mind the more commonly studied "BMO functions").

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    $\begingroup$ @J.W.Tanner Interesting, thanks! $\endgroup$
    – Alp Uzman
    Sep 5, 2022 at 17:45
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    $\begingroup$ Another word related to saltus is somersault . It's not so common nowadays, but saltation can mean the act of leaping, jumping, or dancing. $\endgroup$ Sep 5, 2022 at 20:14
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    $\begingroup$ The word "saltus" was often used in the early 1900s literature in English (but "oscillation" was also used in this literature), and some authors made a distinction whereby one of the words was when the value of the function at the specified point is taken into account and the other word didn't (Hobson does this, for instance). My several comments to History of supremum with parameters gives some information about this topic and where to look. $\endgroup$ Sep 5, 2022 at 20:31
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    $\begingroup$ I looked in some older books I have and it seems "saltus" is less used in the early 1900s than I thought. Blumberg used it in many of his papers and published abstracts and Zbl reviews, which is probably why I thought it was used a lot in the early 1900s. Regarding Sierpinski's 1910 result, I believe this was rediscovered in: Lester Randolph Ford, Pointwise Discontinuous Functions, Master of Arts Thesis (under Earle Raymond Hedrick), University of Missouri (Columbia, Missouri), 1912, iii + 45 + 14 pages (see the last few pages). $\endgroup$ Sep 6, 2022 at 13:39
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    $\begingroup$ @Alp Uzman: For oscillation defined pointwise, 1875 is probably appropriate. For oscillation in an interval, it's not as clear. Hankel certainly dealt with oscillation in an interval in his 1870 memoir, and I think it appears, at least implicitly in Riemann's 1854 thesis on the Riemann integral and Fourier series. And Cauchy dealt with pointwise lim-inf and lim-sup, although probably not their difference (pointwise or in an interval). (continued) $\endgroup$ Sep 6, 2022 at 19:37

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Searching Google Books often leads to useful results. I searched your keywords as "oscillating function" "saltus" Hobson. This leads to the same book "The Theory of Functions of a Real Variable and the Theory of Fourier's Series Volume 1, By Ernest William Hobson, 1921"

On page 284, Hobson has a footnote with a asterisk on oscillation. In other books, Hobson is credited with saltus.

"enter image description here

So it seems Schwingung was translated to oscillation. You will have scavenge German literature. DeepL is an excellent free machine translator. Saltus itself is inspired by German Sprung most likely by the author himself (pg 283).

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    $\begingroup$ I located Pasch' book and ran some OCR. I could not find "Schwingung" (not on page 139 or anywhere), although the more standard "Schwankung" is there (p. 17). Pasch cites Riemann and Luroth for the latter term. $\endgroup$ Sep 6, 2022 at 8:39
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The notation is a bit harsh, but Ascoli defines 'oscillazione' in the following paper (p. 867, paragraph 6; see also p. 869, at the top, for a reference to Riemann, Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe) and formulates a version of the Vitali-Lebesgue theorem using this notion.

Giulio Ascoli, Sul concetto di integrale definito, Atti della Reale Accademia dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali 2 (1875), no. 2, 862-872.

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  • $\begingroup$ It must be added that Ascoli himself complained in Sulla definizione di integrale (1895) that his note, inspired by Riemann's research, had gone completely unnoticed, and that many of his ideas were later attributed to other authors. However, it must be said that the notations used by Ascoli were really too complicated to be appreciated. $\endgroup$
    – user6530
    Jan 30 at 17:48
  • $\begingroup$ The notation is indeed harsh, and I would add that Ascoli's results are not in their final/simplest form. $\endgroup$ Jan 31 at 11:02

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