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::{{ping|D.Lazard}} in my experience this is not the definition of limit commonly used: almost always one takes the hypothesis <math>0 < d(\boldsymbol{x}, \boldsymbol{a})< \delta</math>, specifically removing '''a''' from the ball and allowing the possibility of removable discontinuities. (The article [[limit of a function]] agrees that this is the more common usage.) --[[User:JayBeeEll|JBL]] ([[User_talk:JayBeeEll|talk]]) 16:05, 25 November 2020 (UTC)
:::I have never seen before the definition with <math>0 < d(\boldsymbol{x}, \boldsymbol{a})< \delta,</math> but I must aknowledge that it is used in English Wikipedia. However, in French Wikipedia <math>\boldsymbol{x} = \boldsymbol{a}</math> is not excluded. In German Wikipedia, the Engish definition is given first and is called the "punctured definition"; the French definition is given later in the article, it is presented as "newer" and called "unpunctured". I would guess that the definition used in English Wikipedia is commonly used in US pedagogical mathematics, while the other definition is more commonly used in advanced mathematics. This is only a guess, and would require a verification. As several articles are concerned by this, I'll open a discussion at [[WT:WPM]]. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 17:52, 25 November 2020 (UTC)
== Restriction of ___domain to real variables ==
I am concerned by the redirection of two other pages, relating to multivariate functions and multivariable functions, to this page.
I have no problem or argument with the main content of this page per se (it matches the page name, and I am not qualified to comment on its accuracy or completeness), but the redirection, in my view, equates this important special case, of a "function over several real variables", with the general concept of "a function over several variables".
The general definition of a function over multiple variables does not restrict the function's ___domain to a set of n-tuples of real numbers, nor the range to the reals.
In discrete mathematics it is quite common to define a function over a ___domain which is a Cartesian product: e.g. '''''A''''' X '''''B''''' X '''''C''''', where the sets composed by the product set ('''''A''''', '''''B''''' and '''''C''''' in my example) are simply collections of objects of any kind that makes sense for the problem at hand.
For example, an edge-labelled graph can be defined as follows:
A simple graph '''''G''''' is a pair ('''''V''''', '''''E''''') where '''''V''''' is a set of vertices, and the set of edges '''''E''''' = '''''V''''' X '''''V'''''.
We can say '''''E'''''('''''G''''') to denote the set '''''E''''' in '''''G'''''.
An edge-labelled graph '''''LG''''' is a tuple ('''''G''''', '''''L''''', '''''λ''''') where '''''L''''' is a set of labels, and there is a function '''''λ''''' : '''''E'''''('''''G''''') → '''''L'''''.
The function '''''λ''''' takes as it input a pair of vertices ('''''u''''' ∈ '''''V''''', '''''v''''' ∈ '''''V'''''), and its value is a label '''''l''''' ∈ '''''L'''''.
This is a multivariate function, a function over/of multiple variables, whose value or result is an arbitrary object called a label. There are no real numbers in sight :-).
Such functions commonly appear in computer science as functions over variables of different types with different names, which is just a restatement of the set-theoretic concept of a function whose ___domain is a Cartesian product, since types are sets of values, and we name sets to distinguish them. Once this is made clear, the connection of this page to the page on variadic functions also becomes clearer: they both derive from a more general concept of a function whose ___domain is a product set.
If this is an acceptable framework then I would want to insert some variant of the content just given, with links to this page and to the page on variadic functions, into the page on multivariable functions and redirect the multivariate function page to it (or vice versa).
Apologies for not formatting the mathematical definitions pleasantly or using proper links for the other related pages -- I'm writing in a rush.
[[User:Wikimodoo|Wikimodoo]] ([[User talk:Wikimodoo|talk]]) 07:07, 19 January 2022 (UTC)
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