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: $\displaystyle f^\prime \left({\xi}\right) = \lim_{h \to 0} \frac {f \left({\xi + h}\right) - f \left({\xi}\right)} h$, 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/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)