Talk:Function of a real variable

Latest comment: 6 years ago by D.Lazard in topic Differentiability

Banach spaces

edit

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 (talkcontribs) 19:48, 21 October 2013 (UTC)Reply

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∧Ŝc2ħεИτlk 20:19, 21 October 2013 (UTC)Reply

Continuity and limit

edit

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   where all other definitions I have seen use   . This completely alters the meaning. In particular the following sentence

If   is in the interior of the domain, the limit exists if and only if the function is continuous at  .

is false with the orthodox definition.


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


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)Reply

Differentiability

edit

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)Reply

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)Reply