# Talk:(ε, δ)-definition of limit

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## Notational Issues

I was thinking that this section isn't needed. It just restates the definition of absolute value, and this isn't a section on absolute value. I'll remove it until further discussion 142.151.185.196 (talk) 00:09, 25 September 2008 (UTC)

I don't follow your comment. There is nothing here about absolute values. Katzmik (talk) 08:48, 24 October 2008 (UTC)

Hmm, I was also thinking that this article is sort of already included in the Limit (mathematics) article. Does the epsilon-delta proof really need it's own section? I won't prod it or anything until further discussion but any thoughts would be nice 142.1.130.230 (talk) 16:14, 25 September 2008 (UTC)

The "epsilon-delta definition of limit" is a recognizable term and as such deserves its own page. There are other approaches to the definition of limit. The blanket term limit of a function tends to suggest that this is the only possible approach, which is not the case. An additional point is that the quantifier approach clearly stated at this page applies not only to functions but also to sequences. In short, this page is a different way of slicing the general area of "limits" as compared to the limit of a function/limit of a sequence dichotomy. It is transverse to this dichotomy. Katzmik (talk) 08:52, 24 October 2008 (UTC)

## Why have symbolic definitions at all?

They seem to really hurt the readability to me. Thenub314 (talk) 07:58, 10 October 2008 (UTC)

You are probably referring to the intrinsic difficulty of the epsilon-delta definition, which is related to the logical complexity of working with multiple quantifiers. I think we are all in agreement that the definition is difficult. Generation after generation of calculus students can attest to this fact, and there is no argument about it at all. On the other hand, there are should be a page that presents such a definition clearly so it can be compared to others. Katzmik (talk) 08:54, 24 October 2008 (UTC)

The definition not using formal symbolic notation is no less clear than the symbolic form, and it can easily be compared to other definitions. I was refering to the Manual of style for math articles that recommends avoiding these symbols. (As do countless others on the subject of writing mathematics.) I was not refering to the difficulty of the definition. Thenub314 (talk) 20:44, 26 October 2008 (UTC)

That's an interesting issue. I suggest we discuss it in the context of the new section I just added, comparing the definitions of continuity and uniform continuity in terms of quantifier order. My argument would be that the comparison becomes far more transparent in terms of the standard quantifier notation used in any honors calculus (in addition to many, many other places that go beyond "service" calculus). Katzmik (talk) 08:34, 27 October 2008 (UTC)

I dispute the claim that "standard quantifier notation used in any honors calculus" is used to give the definition of these concepts in every honors course. At none of the universities I have worked at present this notation for the definition. Many follow Spivak's calculus which makes a point of avoiding this notation. And why should this article limit itself to honors students. I first started trying to read about this definition in high school (for example.) In my opinion (at the level of calculus) formal quantifier notation is mainly used as a convenient black board short hand, and not used by more careful teachers. Thenub314 (talk) 21:41, 27 October 2008 (UTC)

We have wiki guidelines that recommend avoiding technical notation and I agree with the general guidelines. In certain situations following them too literally can be an impediment rather than an aid to readability. As I tried to point out before, the comparison of continuity and uniform continuity in terms of quantifier order becomes more lucid if one can use compact notation. It seems hard to argue with this point. Katzmik (talk) 08:32, 28 October 2008 (UTC)

I disagree it becomes more lucid, and I can argue the point. You may simply write out the two statements as sentences, which every one can read regardless if they are familiar with the symbolic notation. The clause about x and delta still change position. Why is this less lucid? Thenub314 (talk) 13:53, 28 October 2008 (UTC)

OK, I hear what you are saying. I happen to think otherwise--perhaps we will have to agree to disagree on this point. If you think this is an important issue, we can ask for some input from other editors. The truth is, I am not sure how far your reasoning against formulas and in favor of sentences would take you. What about Selberg's trace formula for instance? What about Einstein's mass-energy formula? Katzmik (talk) 17:19, 28 October 2008 (UTC)
I have no difficulty with formulas. The examples you gave express relationships between quantities, which is not really what we are talking about. The point is that, as anyone who has tried to read Bourbaki can tell you, overuse of quantifier symbols makes an article very difficult to read. There are certainly places you cannot do without them. But I just don't think they have a place in this article. Thenub314 (talk) 08:47, 29 October 2008 (UTC)
Not everybody will be happy with your distinction between logical formulas and those in other areas of mathematics. At any rate, you should be aware of the fact that you need to argue your case. Katzmik (talk) 10:08, 29 October 2008 (UTC)
I've left some remarks on the same topic (in favour of words, not quantifier symbols) at WT:WPM, with some comments by other editors. As for the Bourbaki remark, it strikes me as odd: their books are essentially (completely outside of Set Theory?) free of quantifier notation, giving instead an excellent example how carefully crafted language and concepts help replace technical formulas (see, e.g., the wordings of definitions in General Topology, the book most relevant for this discussion). Stca74 (talk) 06:32, 30 October 2008 (UTC)
It does seem I have been unfair to Bourbaki. At some point during grad school I had to look at a book that used quantifier symbols at every place it could. I was not able to get very far through it. I thought the book was by Bourbaki, but I must have been mistaken. I read your remarks at WT:WPM, and I agree with them. Thenub314 (talk) 07:54, 30 October 2008 (UTC)
In his new book, Kevin Houston discusses the difficulty of quantifier definitions. I have not read the book yet. I still feel that in the case of a definition involving four quantifiers, having a compact formula expressing the definition can be an aid to understanding, just as in another field of mathematics a formula is an aid to understanding. Clearly, when quantifiers are used as shorthand in sentences so as to reduce ink outlay, such a practice should be eliminated at wiki. Other than that, I am puzzled by what seems to be the anti-quantifier sentiment. Katzmik (talk) 11:44, 30 October 2008 (UTC)

Actually, in my experience, any symbolism is hard for beginners, but often helpful for those who have worked with the subject for a while. (Historically, the former has been a reason for some medieval mathematical text book authors to avoid employing "complicated abstract notation", such as the Indian/Arabic numerals, or Diophantos's symbolic way to present equations.) I've even (admitted rarely) had calculus students who've asked me to translate statements into quantifier dito.

I find it strange completely to avoid the most common compact way to describe the limit definition. There may be reasons to avoid it in the "gentler" article Limit of a function, but if so, they motivate retaining this article as a separate "more advanced" encyclopedic article, with the neat, short, and very common quantifier definitions included. JoergenB (talk) 20:05, 11 November 2008 (UTC)

## Name?

Someone at WP:MATH points out that "The main point of the (ε,_δ)-definition_of_limit is not quite the limit (and its definition), but rather (1) the problem of intuitive understanding of alternating quantifiers, and (2) comparison between analysis and nonstandard analysis." I think as written that seems to be true. If this is the case shouldn't the article be renamed? Thenub314 (talk) 04:28, 28 October 2008 (UTC)

## Category: Calculus

I am not sure why in the list of pages in this category, this particular page appears ABOVE calculus. I thought this was due to the erroneous "|*" that was included at the bottom of the page with the category:calculus, but this phenomenon persists after I removed the |* Could anyone with more technical knowledge help out? Katzmik (talk) 12:37, 30 October 2008 (UTC)

"(" is before "*". I have tried to use " " as the sort key for the primary use of a topic. but it gets killed by the bots. Perhaps if we change the sort key for this article to "ε"? — Arthur Rubin (talk) 20:20, 30 October 2008 (UTC)
Done; set the default-sort for this article to "ε". In Category:Calculus, Calculus is still below List of calculus topics, but we can't have everything. — Arthur Rubin (talk) 20:28, 30 October 2008 (UTC)
Oh, but we can!
As per Wikipedia:Categorization#Typical_sort_keys, eponymous pages have sort key “ ”, while lists have sort key “*”. I’ve made these changes, so now Calculus comes first!
—Nils von Barth (nbarth) (talk) 12:03, 1 May 2009 (UTC)

## d is undefined

In this formula from (ε,_δ)-definition_of_limit#Limit_of_sequence

$L = \lim_{n \to \infty} x_n \Longleftrightarrow \forall \varepsilon>0\;, \exists N \in \mathbb{N}: n>N \implies d(x_n,L)<\varepsilon.\;$

d is undefined. Why not write

$L = \lim_{n \to \infty} x_n \Longleftrightarrow \forall \varepsilon>0\;, \exists N \in \mathbb{N}: n>N \implies |x_n-L|<\varepsilon.\;$

Bo Jacoby (talk) 12:04, 2 November 2008 (UTC).

Agreed. I changed it. Thenub314 (talk) 15:48, 2 November 2008 (UTC)
d(x,y) is a generalised metric, as this definition should work with any function that qualifies as a distance function. Absolute value of a difference is just one specific metric. 62.16.184.248 (talk) 22:04, 25 May 2010 (UTC)

## Multivariable limit definition

Should be considered multivariable limit definition to be added? Examples of onedimensional & multidimensional articles:
-Critical_point_(mathematics)
-Fundamental_theorem_of_calculus

--190.139.10.13 (talk) 22:05, 28 November 2008 (UTC)

## delete merge suggestion?

The merge suggestion listed at the top of the article does not seem to be going anywhere. If nobody objects, I will delete it. Katzmik (talk) 19:59, 29 November 2008 (UTC)

I object. :) Thenub314 (talk) 15:09, 13 January 2009 (UTC)
Overall I think it is better as a separate article as it is notable and there are other definitions of limit of a function. I'm quite willing to be convinced otherwise though. Dmcq (talk) 12:16, 15 December 2011 (UTC)
Well, it seems my edits in this respect were a bit controversial, and I apologize. I was mainly acting on the positive comments from when I brought it up in [math project page]. As I explained there I felt the merge tags were not getting a lot of attention so I wanted to seek a wider opinion before I did anything. The feeling was that the delta-epsilon definition does and probably will always appear in the limit of a function page, as do generalizations and history. When I ask myself what I would want to add to this article that I wouldn't want to add at the other article, I come up blank. So the real concern (to my mind) is a duplication of effort. I take your point that there are other definitions for the limit of a function, but as far as I am aware this is the unique one that has its own page. Thenub314 (talk) 17:28, 15 December 2011 (UTC)

## Slight problem in wording the motivation

"How close is "close enough to c" depends on how close one wants to make ƒ(x) to L. It also of course depends on which function ƒ is and on which number c is. The positive number ε (epsilon) is how close one wants to make ƒ(x) to L; one wants the distance to be no more than ε. "

"no more than epsilon" is less than or equal to epsilon, whereas what is meant (according to the formal definition) is strictly less than epsilon. —Preceding unsigned comment added by Kmddmk (talkcontribs) 02:27, 6 September 2009 (UTC)

During several automated bot runs the following external link was found to be unavailable. Please check if the link is in fact down and fix or remove it in that case!

--JeffGBot (talk) 04:27, 31 May 2011 (UTC)

## Restoring comment.

I feel I should explain my most recent revert. Tkuvho took out some sourced comments saying that they do not belong on this page. But the comments are roughly about the fact that delta and epsilon arguments are implicit in NSA, how does it not belong to the section on NSA on the this page? Thenub314 (talk) 15:28, 14 December 2011 (UTC)

This kind of "implicit" discussion could, perhaps, be mentioned at the page devoted to the book. However, it does not belong in this page as there is no "reception" section here. This comment is simply irrelevant. As I mentioned already, Keisler defines all these concepts without using epsilon, delta techniques. Again: epsilon, delta techniques are not needed to define the basic notions of the calculus in a hyperreal framework. Hrbacek is referring to a separate issue. Tkuvho (talk) 15:31, 14 December 2011 (UTC)
Harbrack's paper is not specifically about keisler's book. He explicitly says that this problem arises in any presentation of NSA. If we are going to have cite a paper that says that in a technical sense the quantifier complexity is reduced, then it seems only appropriate to mention one where the feeling is that in this formulation the usual delta epsilon arguments are in some sense unavoidable. Thenub314 (talk) 16:41, 14 December 2011 (UTC)
Your interpretation of Hrbacek basically amounts to saying that Keisler's book is a fraud, and that contrary to Keisler's claim, one cannot define these notions with infinitesimals. I disagree with your interpretation of Hrbacek. Tkuvho (talk) 16:48, 14 December 2011 (UTC)
If your administrative action was supposed to be a response to this, I must admit it was not very convincing. Tkuvho (talk) 13:04, 15 December 2011 (UTC)

## more on merge

The page serves a different goal as compared to the limit page. The goal is to clarify the logical structure of the epsilon, delta definition. Also a minor goal is to compare it to the infinitesimal definition, about which Hrbacek never claimed that it is a fraud. Tkuvho (talk) 12:31, 15 December 2011 (UTC)

My apologies, given your comments here I thought we agreed on this issue. I am ceasing any further merging activities. Thenub314 (talk) 17:30, 15 December 2011 (UTC)
Well seeing that Hans Adler said fine to the merge I'm more definitely sitting on the fence rather than weak opposing. A good argument would tip me over. Dmcq (talk) 17:46, 15 December 2011 (UTC)
In fact just saw reason above in #delete merge suggestion?. What I think might be reasonable is to keep this page as a redirect pointing to a particular section within the other article as there isn't a lot of it but it is useful to be able to refer to it directly. Dmcq (talk) 17:50, 15 December 2011 (UTC)
This is an excellent idea. But I will still hold off to give more time for comments. Thenub314 (talk) 18:13, 15 December 2011 (UTC)
Are there any objections to merging, and having a redirect to a specific section of the Limit of a function page? Thenub314 (talk) 20:50, 16 December 2011 (UTC)
The reason I object to a redirect is because the title of this page describes more accurately the technique we are dealing with. The notion of limit is present both in the traditional approach, and the infinitesimal approach. The essence of the traditional approach is the epsilon, delta way of expressing the definitions and arguments, not the word "limit". Tkuvho (talk) 13:33, 18 December 2011 (UTC)
Keep in mind the suggestion is not a redirect but a redirect to specific subsection of a page. That subsection could and should describe in detal the technique being dealt with. I agree with you fully, but many parts of this page and the limit page should be common, for example the history of the concept of a limit will necessarily include anything discussed in the history here. The delta epsilon definition is discussed in generous detail all ready at the limit page. And what is worse is that the limit page, which should include other definitions of limits doesn't mention the non-standard definition at all, which is too bad. Thenub314 (talk) 23:33, 22 December 2011 (UTC)
This conversation seems to have stalled, but given the comments supporting a merger here, here and above there seems to be a consensus to merge the two articles. If there are no further objections I will go ahead and do so. Thenub314 (talk) 20:08, 28 December 2011 (UTC)
I believe there are two editors opposed to the merge. Tkuvho (talk) 08:54, 29 December 2011 (UTC)
As I already mentioned, the title of this page reflects more accurately the actual content of the traditional limit concept, so if anything "limit of a function" should be redirected here. As there is little hope of achieving a consensus for such a redirect, I suggest we leave the two pages separate. Tkuvho (talk) 08:56, 29 December 2011 (UTC)
Your correct that there are you and another editor who oppose the merge. I was hoping to convince you, as at one time you seemed to be interested in a merge. Notice that after a merge we could title the new section by the title of this page, and thereby accurately reflect the content. I suppose I see your point about redirecting "Limit of a function" here and the futility of looking for a consensus to do that. But putting aside the issue of page names, which we can always arrange to redirect a reader to the correct place, do you acknowledge there is a duplication of effort between the two pages? Thenub314 (talk) 00:07, 31 December 2011 (UTC)

## try again

In a constructive spirit, Thenub's paraphrase of Hrbacek's comment is misleading. Hrbacek certainly did not mean to say that you can't develop calculus, as Keisler demonstrably does, by using infinitesimal definitions to the exclusion of epsilon, delta clauses. What he meant was that in order to understand how the transfer principle could be correct for the epsilon, delta formula, one needs to convince oneself that it is true also at non-standard points, even though, somewhat paradoxically, microcontinuity may fail there. However, none of this needs to be mentioned in a calculus course, just as equivalence classes of Cauchy sequences need not be mentioned there. Hrbacek would be the first one to be shocked if he discovered that his comment is being interpreted as some kind of refutation of Keisler. Thenub had the humility to acknowledge recently that he is not fully knowledgeable about non-standard analysis; it is time to act on this acknowledgment and refrain from adding misleading material to the page. Tkuvho (talk) 14:04, 15 December 2011 (UTC)

In reply to the comment that I am misrepresenting the source I would like to quote the two parts of his paper that I based the sentences here on:

"At the risk of an overstatement, it is this: while it is undoubtedly possible to do calculus by means of infinitesimals in the Robinsonian framework, it does not seem possible to do calculus only by means of infinitesimals in it. In particular, the promise to replace the ε-δ method by the use of infinitesimals cannot be carried out in full."

"It seems that every attempt to define continuity ultimately has to be grounded on the ε-δ method. As remarked above, the same difficulty appears with derivatives, integrals - in fact, with all standard concepts introduced by nonstandard methods. I see it as a serious problem for the Robinsonian framework, if not as a research tool, surely as a teaching tool and, fundamentally, as a satisfactory answer to the question about the place of infinitesimals, and nonstandard objects in general, in mathematics."

On which I based the sentences: "On the other hand, Hrbacek writes that the definitions of continuity, derivative, and integration in non-standard analysis implicitly must be grounded in the ε-δ method. Thus, from the pedagogical point of view, the hope that non-standard calculus could be done without ε-δ methods can not be realized in full."
I would like to comment further that his concerns are related to things that he feels needs to be mentioned in a non-standard calculus course. (Hence my use of the word pedagogical). The paper specifically focuses on the issue of defining continuity of a standard functions at non-standard points. Though he mentions at least twice that this problem relates to all the basic ideas of calculus. More specifically he writes:

"I realized the crucial importance of this issue for teaching of nonstandard analysis during O'Donovan's talk in Aveiro. While describing his experiences with the nonstandard definition of derivative, O'Donovan recounted some questions his students typically ask: `Can we use this formula when x is not standard? When f is not standard?' The answer of course is NO - but what then are they supposed to use? After all a standard function like $\sin x$ does have a derivative at all x!"

If I am being misleading I fail to see how. Thenub314 (talk) 17:54, 15 December 2011 (UTC)
Here is a sense in which non-standard analysis is "implicitly grounded" in epsilon, delta. To construct the hyperreals, we proceed along the usual track: N to Z to Q to R. We need R first to build R*. How are we supposed to build R first? Well, we need Cauchy sequences of rationals. But how are we going to define a Cauchy sequence? Using an epsilon, delta condition! Only once we have R can we build R*. True. But nobody said we need to build either R or R* in freshman calculus! Certainly a traditional calculus course does not build R. But to interpret Hrbacek's comment as some kind of a refutation of Keisler's approach is to interpret it incorrectly. I keep coming back to the same point: Keisler manifestly does define all basic notions without resorting to epsilon, delta, and he does this in great detail. To take a brief and vague comment by Hrback and interpret it as the discovery of some kind of a tragic flaw is a bit presumptuous. Tkuvho (talk) 20:46, 15 December 2011 (UTC)
Of course there are several constructions of the real numbers, and not all of them involve epsilon's and delta's. Examples I have personally read involve Dedekind Cuts, Fine and consistent families of rational intervals, and formal decimal expansions (which is often mentioned in freshman calculus, though not verified). I am sure there are many others. But that is really beside the point. Now I don't know how I have interpreted this as a discovery, my paraphrasing is just that "Hrbacek writes" not "discovered" or any synonym of the word. This problem is an important part of his paper as it explains why one should consider Stratified analysis. He devotes a section to discussing the problem. He refers to this paper again in his later monthly article. This doesn't strike me as brief or vague. My summary doesn't mention Keisler in any way, or any "tragic flaw". I am simply summarizing what he has written. Thenub314 (talk) 22:47, 15 December 2011 (UTC)
I don't think you are simply summarizing what he is saying. rather, you are providing a WP:Synthesis. As I already mentioned, I think you are misinterpreting his comments. Hrbacek's concern is with foundations of infinitesimal analysis and with subtle theoretical issues that will not be addressed in a calculus course, any more than the construction of the reals. Thus, to understand the transfer principle fully, we need to understand why the first order formula "for every epsilon there is a delta such that, etc." remains true over the hyperreals, as guaranteed by transfer. But the philosophy in the infinitesimal approach is that freshmen do not need to be bothered with this. This attitude prevails whether you are Keisler or Hrbacek.
Notice that your quotes from Hrbacek are still at the "elementary calculus" page, where they can be compared with other reactions and find their proper place. This comment is simply out of place at (ε, δ)-definition of limit. Keisler's paper proves the fact about reducing quantifier complexity in a hyperreal framework. I don't think either of us disagrees with the reliability of this. You might think that building the hyperreal framework in the first place involves complications. This is certainly true, but that's the whole point of the infinitesimal approach. There is no magic wand that will make this material "easy as pie", but the point is to incorporate a large part of the technical difficulties at the foundational level, when developing the (extended) number system. The epsilon, delta gymnastics that some believe to be the essence of analysis can be done once and for all at the foundational level, and the students needn't be "dressed to perform multiple-quantifier logical stunts under pretense of being taught infinitesimal calculus", as a recent source put it. You may or may not agree with this particular point, but that's a separate issue. Hrbacek's comment does not prove or disprove Keisler's result; it is simply unrelated and does not belong on this page. Tkuvho (talk) 09:44, 16 December 2011 (UTC)
I fail to see how the above to sentences constitute a synthesis. Since only one source is involved you presumably mean I am making arguments that are not present in the paper? Let's see if we can find some common ground we can work with. Do you agree with some or any of the following statements?
• Hrbacek writes in the paper cited here that in the NSA approach to calculus the fundamental operations in calculus have "to be grounded on the ε-δ method" and that he sees this as "a serious problem" for the NSA approach to calculus "as a teaching tool".
• He writes in his introduction that "the promise to replace the ε-δ method by the use of infinitesimals cannot be carried out in full."
Given that he repeatedly speaks about the use of NSA as a teaching tool, how do you arrive at the conclusion that he sees a problem with the "foundations of infinitesimal analysis and with subtle theoretical issues that will not be addressed in a calculus course"
And if your correct, and he sees no pedagogical problems, why does he write papers about an alternative approach to teaching using Stratified (or relative) analysis? Thenub314 (talk) 16:25, 16 December 2011 (UTC)
The philosophy of the Nelson-Hrbacek approach is that there shouldn't be two kinds of numbers, making it appear as though infinitesimals and infinite numbers are some kind of imaginary addition after the "true" numbers have been built. They re-build the foundations in such a way that the "hyperreals" become the "reals". They do this by means of adding a unary predicate. You can read more about this at the internal set theory page. What he is getting at is that distinctions between "reals" and "hyperreals" are artificial, and he feels they are confusing to the students. I don't see how he can contest the fact that Keisler defines continuity, derivative, and integral without using epsilontics. It is there in Keisler's book in great detail for everyone to see. Hrbacek's comments are brief and vague; the only time he gets into detail is when he discusses continuity at infinite points. Here I see a major advantage provided by the notion of microcontinuity rather than a disadvantage. Tkuvho (talk) 13:39, 18 December 2011 (UTC)
First, you have not comment directly on the two statements above, nor on the question asked. Second, Nelson-Hrbacek do not both add a unary predicate, the point of Hrbacek's approach in this paper is to add a binary predicate. Also, I am familiar with the Internal Set Theory page. While your statements about Hrbacek's philosophy are interesting, they are a bit of a tangent. I find his comments neither brief nor vague, as he does discuss how these issues are addressed by Robinson, Neloson framework, and how he would attempt to handle them within NSA, then discusses what can be done. This goes on for several pages before he finally leaves the topic and moves on to describing Stratified analysis. For anyone interested they may also want to take a look at Talk:Elementary_Calculus:_An_Infinitesimal_Approach#Synthesis. Thenub314 (talk) 21:09, 18 December 2011 (UTC)
I tend to think Hrbacek's comment is mostly foundational. The fact that he uses terms such as "ultimately" and/or "implicitly" (the latter term may be your paraphrase, I can't recall) indicates that his claim needs to be suitably interpreted. I think if we include a reference to Keisler's infinitesimal definitions, this point will be clear to the reader, as well. Tkuvho (talk) 12:55, 19 December 2011 (UTC)
As I point out below, I don't think Keisler's definitions are actually a response to Hrbacek's argument, as Hrbacek is objectively correct when he points out that Keisler's definitions do not cover non-standard values of the input. Does Keisler ever explicitly say why it is reasonable to view the physical line as the hyper-real line but only define continuity and derivatives at standard values? That sort of comment by Keisler would be a useful thing to add to the article, if it exists. But I have not found it in Keisler's book myself. — Carl (CBM · talk) 15:31, 19 December 2011 (UTC)
As I mentioned elsewhere, both the real line and the hyperreal line are idealisations, but as far as applications in physics are concerned we mostly care about values of real functions at real points. Tkuvho (talk) 08:58, 29 December 2011 (UTC)

## Deletion

The page numbers I provided for the infinitesimal definitions of continuity, derivative, and integral in Keisler's book have recently been deleted. Their inclusion is appropriate to counter the impression that may be given by the earlier citation from Hrbacek that such definitions are not available. Namely, the paraphrase of Hrbacek that has been recently added to the page is worded in such a way as to suggest that such definitions are not provided. The page numbers correct such a misperception. Tkuvho (talk) 15:21, 19 December 2011 (UTC)

(I did not remove the paragraph.) One key issue is that Hrbacek is talking about definitions for arbitrary values of the input, including non-standard ones, while Keisler is only giving a definition that works for standard values. Hrbacek's argument seems to be accurately represented here. I had even added "Hrbacek argues" to the sentence to make it clear that this is Hrbacek's argument. Could you explain what you find "dubious"? — Carl (CBM · talk) 15:25, 19 December 2011 (UTC)
It would be helpful to emphasize a bit more that the explicit, epsilon-free definitions are provided, by citing the page numbers, so as not to mislead the reader. After all, most people are interested in real functions and real values, whether input or output. I find Hrbacek's criticism of limited significance for someone interested in calculus of real functions, but at any rate if we include this we should make sure the reader realizes that the real part of the picture is covered satisfactorily in Keisler's book. Tkuvho (talk) 15:37, 19 December 2011 (UTC)
Wher in Keisler's book is there an explicit, epsilon-free definition of the derivative at a nonstandard value of the input? I have access to the online version on Keisler's web site. The definition on page 45, as I have pointed out elsewhere, does not respond to Hrbacek's criticism, in fact that definition is the exact target of the criticism. — Carl (CBM · talk) 15:43, 19 December 2011 (UTC)
At the risk of being repetitive, I will mention that Keisler does not provide such a definition on page 45. The "extension principle" to the effect that every real function has a hyperreal extension is part of the foundational package assumed given in the hyperreal approach, just as a coherent system of the real numbers is assumed in the calculus: first year calculus proves the intermediate value theorem, but does not construct the number system where this theorem is valid). Applying the extension principle to the function g=f', we get its values at non-standard points. It seems reasonable to include the page numbers in both articles rather than just one, if we include the criticism in both articles. Tkuvho (talk) 15:48, 19 December 2011 (UTC)
The issue is that I don't see that that page numbers are actually a response to Hrbacek's criticism. They only give examples of the thing Hrbacek is criticizing, namely that the definitions are only valid for standard points, and that as literally written the definitions do not transfer and cannot be used to compute derivatives at non-standard inputs. — Carl (CBM · talk) 15:59, 19 December 2011 (UTC)
The page numbers were merely an attempt to provide a sourced statement to the effect that such definitions are provided, so Hrbacek's criticism won't mislead the reader. Tkuvho (talk) 16:13, 19 December 2011 (UTC)
But the page numbers do not actually give a response, because they don't do what Hrbacek says is impossible, namely give a definition for nonstandard inputs. It's like if I said that the real-valued definition of square root in some book does not handle negative inputs, and someone gave a reference to the page number in the book where the real-valued definition. What would be useful is an actual reference where Keisler explains why, even if the physical line is hyper-real, we would want to do calculus just with standard reals. — Carl (CBM · talk) 17:09, 19 December 2011 (UTC)
Your analogy with roots of negative numbers is not really to the point since everybody knows that (traditional) calculus deals with real objects. I am not sure why one would a priori expect things to be any different for the infinitesimal approach. The hyperreals are a tool for studying real objects. As far as your comment on the physical line, I responded at talk:Non-standard calculus, perhaps we can amalgamate the discussion there. Tkuvho (talk) 17:17, 19 December 2011 (UTC)

I have a suggestion on the readability of the paragraph that explains the epsilon-delta concept, which currently seems a bit robotic. Here is the original -> 'How close is "close enough to c" depends on how close one wants to make ƒ(x) to L. It also of course depends on which function ƒ is and on which number c is. The positive number ε (epsilon) is how close one wants to make ƒ(x) to L; one wants the distance to be less than ε. The positive number δ is how close one will make x to c; if the distance from x to c is less than δ (but not zero), then the distance from ƒ(x) to L will be less than ε. Thus δ depends on ε.'

The minor amendments would read -> 'How close is "close enough to c" depends on how close one wants to make ƒ(x) to L. It also of course depends on which function ƒ is and on which number c is. [Let] the positive number ε (epsilon) [be] how close one wants to make ƒ(x) to L; [thus] one wants the distance to be less than ε. [Further, if] the positive number δ is the how close one will make x to c[,] [and] if the distance from x to c is less than δ (but not zero), then the distance from ƒ(x) to L will be less than ε. [Therefore] δ depends on ε.' SamCardioNgo (talk) 16:02, 30 October 2012 (UTC)

I agree that this discussion could be improved. Give it a try, by all means. Tkuvho (talk) 10:36, 31 October 2012 (UTC)

## Minor Correction

This is a much smaller suggestion re the following line 'Hrbacek writes that the definitions of continuity, derivative, and integration in non-standard analysis implicitly must be grounded in the ε-δ method in order to cover also non-standard values of the input.' - should the [also] not be [all]? SamCardioNgo (talk) 16:16, 30 October 2012 (UTC)

Since it isn't really clear what he means the best would be to provide a direct quotation if possible. Tkuvho (talk) 10:36, 31 October 2012 (UTC)

## The 1st figure is misleading

... since, in general cases, the preimages of L+epsilon and L-epsilon do not correspond, necessarily, with c+delta and c-delta respectively. kmath (talk) 20:00, 26 June 2013 (UTC)

Here [1] there is a realistic situation depicting the quadratickmath (talk) 20:04, 26 June 2013 (UTC)

## Symbolic Notational Issue

There is a slight inconsistency in this: When do you use the colon and when not? Instead of $\forall \varepsilon > 0\ \exists \ \delta > 0 : \forall x\ (0 < |x - c | < \delta \ \Rightarrow \ |f(x) - L| < \varepsilon).$

I would prefer at the least a systematic

$\forall \varepsilon > 0:\ \exists \delta > 0:\ \forall x:\ (0 < |x - c | < \delta \ \Rightarrow \ |f(x) - L| < \varepsilon)$

Or even more unambiguously

$(\forall \varepsilon > 0:\ (\exists \delta > 0:\ (\forall x:\ (0 < |x - c | < \delta \ \Rightarrow \ |f(x) - L| < \varepsilon))))$

Ok given that some people here dont want symbolic at all, I guess the 3rd wont fly but can we have at least the second?

Rpm13 (talk) 17:55, 12 March 2014 (UTC)