# History of supremum with parameters

I had the following 'history of mathematics' question:

Who first used the notion of supremum explicitly involving parameters?

Let me provide a positive example of the latter notion:

Baire defines 𝑀(𝑓,𝑎,𝑏) as sup𝑥∈[𝑎,𝑏]𝑓(𝑥) in his 'Lecons sur les fonctions discontinues'. Clearly, this supremum involves parameters, as given by the variables $$f, a, b$$ in 𝑀.

Negative results can be found in the work of Weierstrass and Darboux: the former merely describes the supremum, while the latter does use the notion sup𝑥∈[𝑎,𝑏]𝑓(𝑥), but nothing like Baire's function 𝑀.

As a generalisation of my question, I am interested in any pre-1900 author using $$`$$functional' notation akin to Baire's function 𝑀.

• To what extent have you looked through the works and treatises by well known people such as Borel, Dini, Jordan, Darboux, Hankel (his 1870 memoir), Vitali, Peano, du Bois-Reymond, Harnack, Ascoli, Pringsheim, Thomae, Stolz, etc.? Essentially all of their publications are feely available on the internet -- just google ("regular" google and google-books) the title (or a few title words) along with last name, and add "archive.org" as a keyword if you want items collected there (although archive.org is mostly for actual books and not journal volumes). Nov 28 '21 at 7:18
• So e.g. Hankel (1870) and Dini still use very "descriptive" language: they do define the oscillation of a function $f$ and similar constructs, but do not introduce an 'oscillation function' $\omega_{f}(x)$ as we would today. Darboux (1875, memoir on discontinuous functions) does write $\sup_{x \in [a,b]}f(x)$, but introduces nothing like Baire's function $M(f, a, b)$. I would say the same for Vitali and Jordan. Nov 28 '21 at 9:22
• I've found that tracking down notation originations to be VERY difficult in general, as secondary (historical) sources rarely pay attention to it (various vector and quaternion notations, mostly in flux in the 1880s to a few years after 1900, is a rare exception) and primary sources often introduce/modify notation without comment. Regarding oscillation, I believe Ascoli (1875) was the first to introduce the notion of oscillation at at point (paper with usable URL is [1] in this answer; (continued) Nov 28 '21 at 15:34
• see also this 31 January 2003 sci.math post; for more recent and precise results involving oscillation, but probably not helpful for your specific question here, see this answer). For a very detailed survey of Baire's work, see Notes et documents sur la vie et l'oeuvre de René Baire by Dugac (1976). (continued) Nov 28 '21 at 15:44
• Also of possible use is: Giorgio Letta, Le condizioni di Riemann per l'integrabilità e il loro influsso sulla nascita del concetto di misura [The Riemann conditions for integrability and their influence on the notion of measure], Rendiconti Accademia Nazionale delle Scienze detta dei XL. Memorie di Matematica (5) 18 (1994), pp. 143-169. MR 96g:01026; Zbl 852.28001 Incidentally, I hired a native Italian speaker (non-mathematician) to help me translate Letta's paper (continued) Nov 28 '21 at 15:44