I had the following historical question (with implications for logic/reverse math):
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 $\sup_{x\in [a,b]}f(x)$ in his 'Lecons sur les fonctions discontinues'. Clearly, this supremum involves parameters, as given by the variables 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 $\sup_{x\in [a,b]} f(x)$, but nothing like Baire's function $M$.
My question is motivated by the observation that Baire's construct yields the Suslin functional (and hence $\Pi_1^1$-comprehension).
As a generalisation of my question, I am interested in any pre-1900 author using `functional' notation akin to Baire's function $M$.
