Talk:Epsilon-Function Differentiability Condition

Perhaps this page and Characterization of Differentiability this one should be both transcluded into one page, or are they distinct enough to be kept seperate? --GFauxPas (talk) 19:27, 31 December 2012 (UTC)

They could not be more different. --prime mover (talk) 19:58, 31 December 2012 (UTC)

This theorem references differentiability in $\C$, not $\R^n$. Apart from that, they say the same thing, except that this version of the theorem does not use the $\Delta$ notation. The reasons I've uploaded a different version are: 1) It's the version that my source uses to prove theorems. 2) I needed a version that references $\C$.

prime mover, I don't think we should interpret this theorem as a second definition of differentiablity. My source uses this theorem as a tool to make certain proofs easier, simply because the proofs do not have to use the usual fraction: $\ds f' \left({\xi}\right) = \lim_{h \to 0} \frac {f \left({\xi + h}\right) - f \left({\xi}\right)} h$, and so avoid the case where $h=0$.

However, I need a version of this theorem that also works in $\R$, like the one-dimensional version of Characterization of Differentiability. The proof will be completely similar to this proof, except using $\R$ in place of $\C$. Should I upload a new theorem, or may I simply amend this theorem so it adresses both $\R$ and $\C$ simultaneously? --Anghel (talk) 12:13, 1 January 2013 (UTC)

Strikes me that it ought to be possible to prove this very same concept in a general metric space, and also link this with the concept (not sure how it's formulated, it's beyond my current level of attainment) of the general topological space (although in this context there are limitations to manifolds, IIRC).

If you are able to gather all the strands together into something shaped like Definition:Continuity we might be going somewhere but as I have no idea of the technical details of the source work you are using I don't know how this would fit together.

I do note that the work that sources this entry seems to define the derivative using different language and terminology from what is used on this site (which is conventional for mid-20th-century mathematical works in English) - you might want to revisit the introductory chapters of this work, and introduce the notational strategy into the existing pages on this site, rather than jump into the middle of the book. The latter technique can lead to inconsistent threads of thought when linking back to more basic concepts. --prime mover (talk) 12:30, 1 January 2013 (UTC)

Being curiuos, what exactly is the difference in language and terminology that you refer to? My source uses as the definition of continuity: "$f$ is continuous at $z$ if $z \to z_0 \implies f(z) \to f(z_0)$", or alternatively the $\epsilon-\Delta$ definition. My source also uses the same limits of fractions to define differentiabilty as you have on ProofWiki. I'm sorry for asking, but I really want to follow the existing conventions on your site. Bear in mind that English is not my first language, so any differences in terminology may be due to my bad choice of words.

I don't think the differentiablity condition in this theorem extends to general metric spaces, as the condition requires multiplication with $h$. However, I can make the theorem work for $\R$. I think I'll generalize the theorem to $\R$, and then make two slightly different proofs - one for $\R$ and one for $\C$. Unless you have a different suggestion. --Anghel (talk) 13:53, 1 January 2013 (UTC)

The difference is that the derivative is given in terms of the $o \left({h}\right)$ operator. That's what I was talking about.

There should already be such a condition somewhere in the definitions of differentiability for real numbers. I haven't checked for a while.

The use of open balls led me to think of metric spaces. --prime mover (talk) 15:50, 1 January 2013 (UTC)

I rewrote the theorem so it works for $\R$ too.

Definition:Differentiable Real-Valued Function at Point is the condition you were looking for. Maybe I should add a link from that definition to this page, so we can see that the two definitions are equal. --Anghel (talk) 23:03, 1 January 2013 (UTC)

Restructure?

This is one of those cases where a two-part proof has more than one proof. There are two "necessary condition" proofs but only one "sufficient condition" proof.

In these circumstances, it is arguably better to structure it with a page called Alternative Differentiability Condition/Necessary Condition with two proof subpages Alternative Differentiability Condition/Necessary Condition/Proof 1 and Alternative Differentiability Condition/Necessary Condition/Proof 2 along with another page Alternative Differentiability Condition/Sufficient Condition. Thus there is no need to force the proof to fit the structure of having to find two proofs for the sufficient condition where there is but one.

There is an existing proof somewhere which uses this paradigm - not sure where it is, may be around the Heine-Borel Theorem or something complicated like that.

Also suggest a change of name from "Alternative Differentiability Condition" - we might want to think about going down the road of e.g. "Differentiability Condition 1" and "Differentiability Condition 2", so as to provide an equal level of service, so to speak, and also to allow for a smoother migration to "Differentiability Condition 3" etc. as they approach. --prime mover (talk) 21:08, 10 January 2013 (UTC)

It's true that the wordings of the Sufficient Conditions are identical, but the links are different: Proof 1 links definition of differentiability to Definition:Differentiable Real Function, while Proof 2 links definition of differentiability to Definition:Complex-Differentiable Function. Of course, we could change that to a link to Definition:Differentiable Mapping

As for the various differentiability conditions, I think there should be only one main differentiability condition, and that's the one I've linked to. If we should rename this theorem, I'd suggest a descriptive name instead of a number. Something like "Taylor-Polynomial Differentiability Condition", maybe? --Anghel (talk) 21:26, 10 January 2013 (UTC)

If there are two different equivalent definitions for differentiability, they should both be included as equivalent definitions - that is the direction in which this site is currently evolving. This is especially the case if the alternative definition is used by some sources as the prime definition, with our "main" definition included by that source as its equivalent "secondary" one.

As for the use of a descriptive name rather than a number, the jury is out on that - both have advantages, and the "numbering system" is currently what is being implemented as standard here. (For a start it makes it unnecessary to invent a name which you may have to strain for.) That's not the important point: the important point is that there are equivalent definitions each of which should be allowed full definition status.

Besides, your structuring of the proofs is contrary to your professed preference - they are indicated by number whereas in fact their names are what is relevant. Now I look at them closely, they are not equivalent - they actually prove soemthing different. One is a proof on the real number line, the other on the complex plane. As they are not proving the same thing, they should not even be on the same page. One should be on a subpage of the main theorem specifically referring to real numbers, the other should be on a different subpage which specifically refers to complex numbers. As such, it is the theorem page itself which needs to be split, not just the proof. --prime mover (talk) 21:39, 10 January 2013 (UTC)

In my book, numbered pages should only occur as subpages of properly named pages. Currently this condition isn't violated anywhere, and I'd like to continue that status. --Lord_Farin (talk) 22:01, 10 January 2013 (UTC)

Renaming

My suggested improved name for this lemma: Epsilon-Function Differentiability Condition Anghel (talk) 15:23, 21 August 2022 (UTC)

Yes okay, feel free to run with that --prime mover (talk) 15:32, 21 August 2022 (UTC)

Job done -- please feel free to check that the various instantiations of the links to the new pages point towards the correct page. Thanks --prime mover (talk) 05:54, 22 August 2022 (UTC)

I am in the process of checking these page, of which the differentiability rules need to be merged. Anghel (talk) 14:27, 22 August 2022 (UTC)