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.