Talk:Function of a real variable

Banach spaces
Should we mention that most of these things (limits, derivatives, etc.) can be performed in Banach spaces? What about absolute continuity? — Preceding unsigned comment added by Brirush (talk • contribs) 19:48, 21 October 2013 (UTC)


 * Yes, should include absolute continuity.
 * About Banach spaces: Possibly, near the end.
 * I'm not familiar with Banach spaces, so please feel free to keep up the good work! Thanks for expanding this article. For now I just used up my free time editing this article... M&and;Ŝc2ħεИτlk 20:19, 21 October 2013 (UTC)

Continuity and limit
You wrote "Until the second part of 19th century, only continuous functions were considered by mathematicians." I don't believe it. For example https://en.wikipedia.org/wiki/Floor_and_ceiling_functions states "Carl Friedrich Gauss introduced the square bracket notation [ x] for the floor function in his third proof of quadratic reciprocity (1808)."

More to the point, in the definition of limit you use the condition $$d(x, a)< \delta$$ where all other definitions I have seen use $$0<d(x, a)< \delta$$. This completely alters the meaning. In particular the following sentence "If $a$ is in the interior of the domain, the limit exists if and only if the function is continuous at $a$."is false with the orthodox definition.

With the orthodox definition, any real number would pass as a limit at the isolated points of $$X$$ if these were included as candidates for $$a$$ so the range of candidates is normally taken as the set of limit points of $$X$$ rather than the topological closure of $$X$$.

Also the restriction of points of continuity to the interior of the function's domain seems unnecessary and, I think, also unorthodox. In Apostol's "Mathematical Analysis", for example, a function can be continuous at any point of the domain - he specifically mentions that a function is continuous at any isolated point in its domain. Martin Rattigan (talk) 02:21, 30 April 2015 (UTC)

Differentiability
Two issues: is there a difference between "derivable" and "differentiable"? It seems like "derivable" is just a seldom used synonym, unless I am misunderstanding this sentence. Also "It is often implicitly assumed that the function is derivable in some interval." Really? By whom? 67.186.58.77 (talk) 07:38, 10 May 2018 (UTC) Alsosaid1987 (talk) 07:39, 10 May 2018 (UTC)
 * I agree that it is better to replace "derivable" by "differentiable". However the words are not exactly synonym. One means "having a derivative", the other means "having a differential". This is the same in the case of one variable, but not in general. I'll try to find a better formulation for avoiding "assuming". D.Lazard (talk) 08:40, 10 May 2018 (UTC)