Talk:Uniform continuity

message: This page doesn't need math mode, for anyone with a Unicode-compatible browser.
We should consider making the math renderer produce HTML for simple expressions like this that can trivially be rendered as HTML + Unicode.
:Yes, please, in my browser the equations are about twice the font size of the rest of the article and it's quite jarring. I don't see any way to overcome this using PNG equations, since there's no way to know how any given browser will render the text. But as another option, how about a perference setting that causes equations to be displayed in MathML instead? If a browser knows how to handle MathML, then it probably knows how to make it look good as well. Bryan
::We're not XML-clean yet, so we can't embed MathML in our output stream and expect anything to be able to use it. --Brion 08:45 Jan 7, 2003 (UTC)
The "In other words, the Slope is Bounded" comment is false. sqrt(x) is uniformily continuous on [0, infinity) but the slope can be made arbitrarily large by looking at points near 0.
\frac{\sqrt{x}-\sqrt{0}}{x - 0} = \frac{1}{\sqrt{x}} 
so i'm taking down the "slope is bounded" comment.
AlfredR 05:00, 6 December 2006 (UTC)

message: The introduction stated that continuous functions on a closed interval are uniformly continuous. We need compactness here, as the example f:\mathbb{R}\rightarrow\mathbb{R}, f(x)=x^2 shows: Choose \epsilon=1, let \delta>0 and set y=x-\delta/2. Then |x-y|=\delta/2<\delta, but |x^2-y^2| = |x-y| |x+y| = \delta/2 |2x-\delta/2| \rightarrow \infty as x \rightarrow \infty. --Lacce (talk) 13:45, 22 December 2007 (UTC)
thread_title:  Continuous functions on a closed real interval 

message: The lead says "furthermore the size of the changes in f(x) depends only on the size of the changes in x but not on x itself". I know this is meant to be an informal statement, so maybe I should not be too picky, but I find this somewhat misleading. The changes in f(x) do depend on x (as well as on the size of the changes to x) but they can be bounded in a way that only depends on the size of the changes. Would saying that make the lead too technical? (I think anybody wanting to learn about uniform continuity will have to be prepared to encounter some technicality anyway.) Marc van Leeuwen (talk) 16:34, 17 May 2008 (UTC)
thread_title:  misleading lead? 

message: I made some substantial changes, mostly additions, in the section on "properties" and also added a section on the relation between uniform continuity and the extension problem.  
Further edits would still be desirable.  (In particular, I find the lead unsatisfactory, but didn't want to start there.)  I took out the bit about nonstandard calculus and uniform continuity, but anyone who wants to add it back will find that I left an appropriate place for it in the exposition.  Plclark (talk) 22:56, 7 September 2008 (UTC)
thread_title:  Some revisions/additions 

message: In response to Plclark's previous comment, please let me know what you meant when you introduced the discussion of local versus global in the context of uniform continuity.  Apparently you meant that in order to define continuity, it suffices to use a single point x, whereas to define uniform continuity, one needs to refer to pairs.  In other words, in the standard approach, the property cannot be defined pointwise.  Is that the gist of what you meant to say? Katzmik (talk) 14:34, 27 October 2008 (UTC)
thread_title:  local versus global 

message: In response to the 7-step argument above, the first six are of course correct :)  The correctness of the last step depends on how you define "local", see previous section and non-standard calculus#uniform continuity.  I perfectly admit to the possibility that I am in error in which case I would be happy if this wiki discussion clarifies my thinking, as I am an expert neither in logic nor in non-standard analysis (and would like to overcome such a shortcoming).  Katzmik (talk) 14:39, 27 October 2008 (UTC)
replies: text: OK, I think I have understood the issue now. While I don't think non-standard analysis should be mentioned here, I would have been content with a compromise consisting of a single sentence. But it turns out the matter is so complicated (don't forget that we are at an article about advanced school-level and standard first year university level maths) that we would need several sentences to explain what the statement means, and probably still risk that people misunderstand it. --Hans Adler (talk) 19:19, 27 October 2008 (UTC)
replies: text: One of the editors raised an interesting issue of the possiblity of a standard pointwise definition of uniform continuity in an extended domain.  If this can be done, obviously that changes the picture altogether.  If not, perhaps we can talk about devising such a sentence, which in my mind should not be difficult.  Katzmik (talk) 08:59, 28 October 2008 (UTC)
thread_title:  7-step argument 

message: I added a comment about the term "pointwise" without changing any substantive material here.  Let's agree about the mathematics first before making any substantive changes.  Katzmik (talk) 09:22, 28 October 2008 (UTC)
replies: text: I am not sure that I can agree with using the term "pointwise" in this article. To me it sounds way too strong: To see whether two functions agree on a domain, it's enough to compare them pointwise, but to see whether a function is continuous we need to check an open neighbourhood of every point. Even though some people may use the term in a loose sense that covers this (I am not aware that anyone does), I think to minimise the potential of confusion we shouldn't do it. --Hans Adler (talk) 11:16, 28 October 2008 (UTC)
replies: text: Oops, OK.  I thought this was one uncontroversial edit.  No problem, we can take it out.  What about "pointwise germ" or something of that sort?  At any rate, it would be interesting to solve the riddle posed above about a possible pointwise germ standard definition of uniform continuity.  Katzmik (talk) 11:19, 28 October 2008 (UTC)

text: Katzmik, I realise you think I am an ignoramous with a complete lack of understanding, but perhaps you could humour me by explaining just what you mean by "pointwise" here. You say above "the non-standard definition is a pointwise definition and therefore is more accessible than the standard definition which necessarily appeals to pairs of points" - but the definition of uniform continuity in non-standard calculus says "for every pair of hyperreals x and y in I*, if x\simeq y then f^*(x)\simeq f^*(y)". Can you explain precisely what is the "pointwise" property that the non-standard definition has and the standard definition lacks ? Gandalf61 (talk) 11:30, 28 October 2008 (UTC)
replies: text: Hi Gandalf61,  Thanks for your comment.  I am sorry I used the colorful language of "motherhood and apple pie" in an earlier discussion, but I think we should try to get over this minor episode.  At any rate I already apologized at your talk page as I recall, and I do so again to dispell any lingering doubts.  I don't happen to think you are an "ignoramus with a complete lack of understanding", either.  You asked a series of good (if somewhat naive) questions at the talk page at non-standard calculus, and I would like to remind you that I took a considerable amount of time to answer (most of) them.  To answer your specific question, the answer has to do with the order of quantifiers.  Let me state the definition in a more detailed fashion that would make the point clear:  f will be uniformly continuous if for every x in I*, whenever y\sim x, one also has f(y)\sim f(x).  Here the star above the I is the key.  I would be glad to provide further clarification if necessary.  Katzmik (talk) 17:30, 28 October 2008 (UTC)
replies: text: Well, yes, further clarification is necessary, because I don't understand what you mean by "order of quantifiers" and I can't see how moving from the interval I to its extension I* introduces some "pointwise" property into the non-standard definition of uniform continuity that the standard definition lacks. It might help if you explained precisely what you mean by "pointwise" and/or "order of quantifiers". Gandalf61 (talk) 13:42, 29 October 2008 (UTC)
replies: text: I am sorry I now realize my previous comment was too vague (I was sort of assuming you read the subsection Uniform continuity).  At any rate, let's stick for the moment to the standard domain.  What is the difference between ordinary continuity and uniform continuity?  It is certainly the order of the quantifiers.  Namely, in ordinary continuity we choose x before we choose δ, whereas in uniform continuity it is the opposite.  Now if we apply the the definition of ordinary continuity, but on the natural extension I* instead of I, the result turns out to be equivalent to uniform continuity of f.  In other words, we still work with the pointwise germ of the function, but this germ is examined at some non-standard points in addition to the standard points.  Perhaps some of these explanations could be added at non-standard calculus, please feel free to edit (I noticed you have made some edits there before).  Katzmik (talk) 13:53, 29 October 2008 (UTC)
replies: text: Hmmm. You said "if we apply the the definition of ordinary continuity, but on the natural extension I* instead of I, the result turns out to be equivalent to uniform continuity of f". Equivalent how ? Does that mean that any function that satisfies the non-standard definition of "ordinary" continuity everywhere in an interval I* also satisfies the non-standard definition of uniform continuity in I* ? Gandalf61 (talk) 14:44, 29 October 2008 (UTC)
replies: text: I am sorry I am not sure what you are asking, could you clarify?  Are you asking what happens when we look at the hyperreal analog of the standard definition of uniform continuity, instead of the definition that appears in Non-standard calculus#Uniform continuity? Katzmik (talk) 15:20, 29 October 2008 (UTC)
replies: text: Oh, I give up. I was simply trying to understand what you meant by "pointwise", but this discussion has now become "pointless". Gandalf61 (talk) 15:32, 29 October 2008 (UTC)
replies: text: That particular question is not so mysterious, please see the section on the lack of uniform continuity of x2, which is due to lack of ordinary continuity at a single infinite point, as explained there.  That's what happens in general, as well.  I am sorry I did not understand your question; there are several simultaneous discussions going on, at this page alone, and I was getting confused. Katzmik (talk) 15:42, 29 October 2008 (UTC)
thread_title:  Pointwise 

message: As far as I could see, neither of the references given at the end of nonstandard calculus discusses  uniform continuity.  
::It's in Keisler's "foundations of infinitesimal calculus", page 45. Katzmik (talk) 10:35, 29 October 2008 (UTC)
replies: text: Thanks for adding this reference.  (I was initially confused by the fact that the link itself was to Keisler's "Elementary calculus", but I fixed this.)  Note that this source does not describe nonstandard uniform continuity as being "local" or "pointwise".  It would still be very useful to find a primary source which does so. Plclark (talk) 15:17, 29 October 2008 (UTC)
replies: text: This certainly true :) and should be mentioned.  Katzmik (talk) 10:36, 29 October 2008 (UTC)

text: If we have reached a consensus that standard completion of R would not do, we can proceed further.  Katzmik (talk) 10:37, 29 October 2008 (UTC)
thread_title:  Comment about nonstandard uniform continuity 

message: Is this discussion on non-standard analysis not getting slightly out of proportion in terms of its significance to the concept of uniform continuity? First of all, the usual definition is simple enough: f is uniformly continuous iff in order to guarantee that f(x) and f(y) are V-close for some "measure V" of closeness, it is enough to ask that x and y are U-close, for suitable U. Supply distances for U and V into this general definition for uniform spaces, and the resulting standard definition for metric spaces is hardly difficult to grasp. I find it hard to see how this would be conceptually more taxing than the transfer principle required to really understand the non-standard definition, as I argue below.
:The difficulty of definitions involving multiple quantifiers is not my opinion, but rather a phenomenon documented in the literature. Katzmik (talk) 10:51, 30 October 2008 (UTC)
replies: text: True, but I think the point is understanding why there exists a model for Robinson's axioms requires more mathematical sophistication then understanding two quantifiers.  Well ok, I think I am not quite hitting the point, Stca74 is making the point that understanding the transfer principle is more difficult then two quantifiers.  Thenub314 (talk) 15:43, 30 October 2008 (UTC)

text: I think it is possible and has been done.  I would mention a paper by one of the most talented topologists active today, namely Shmuel Weinberger.  His work is mostly "standard" but he recently wrote a paper using non-standard techniques.  The reference is at his page as well as at Non-standard analysis. Katzmik (talk) 10:51, 30 October 2008 (UTC)
replies: text:  I took a glance through Shmuel's paper, and it certainly doesn't directly answer this question.  Maybe it can be generalized, maybe it cannot.  If it can we should include an actual reference.  On a separate subject, between me and Katzmik, Shmuel uses the terms "monad" in the sense I introduced it, on page 3. Thenub314 (talk) 15:43, 30 October 2008 (UTC)

text: The pedagogical benefits (based not not speculation but rather classroom experience) were detailed in a brochure Keisler wrote a couple of decades ago, partly as a response to Bishop's attack on him.  I would mention at least two very significant advantages: 1. reduction of quantifier complexity, already discussed; 2. replacement of "inverse problem" by a "direct problem".  You may recall from your calculus days the way you went through the epsilon, delta exercise, obtained a certain expression in terms of epsilon as a bound for delta, and them went on to invert it so as to require an odd-looking bound for epsilon so as to end up with "delta" on the nose in the end.  Now we all enjoy this stuff enormously, but some calculus students could argue that it is an acquired taste. Katzmik (talk) 10:51, 30 October 2008 (UTC)
replies: text:  It should be noted that, despite Keisler's brochure, it does not seem to be a very popular program.  Are there any curriculums using his text for introductory calculus? Are there independent verifications of it's pedagogical benefit? Thenub314 (talk) 15:43, 30 October 2008 (UTC)

text: I am sorry I did not follow the last paragraph very well.  At any rate, the current state of uniform continuity is fine as far as I am concerned.  If you think this is an important point we can discuss it elsewhere.  Katzmik (talk) 10:54, 30 October 2008 (UTC)
replies: text:  I do think there is undue weight.  The examples given in the non-standard section do lend much enlightenment, prehaps we could delete them. Thenub314 (talk) 15:43, 30 October 2008 (UTC)
thread_title: Out of proportion

message: Your comment "in the actual real line there are no infinitesimals at all, and so infinitesimals don't play any role in most mathematicians' understanding of real analysis" deserves a separate section, as well.  What do you mean by the adjective "real" when you say "the actual real line", as well as of the related quote from Keisler at non-standard calculus and/or Bishop-Keisler controversy? Katzmik (talk) 16:14, 29 October 2008 (UTC)
replies: text: I mean the object known as the "real line". Keisler is entitled to his own opinions, of course, but every exposition of nonstandard analysis I have seen begins with the premise that R* is not the same as R, but is an extension of it. On the other hand, in order for R to be the completion of Q, R has to be Archimedean. I'm not sure this conversation is still relevant to the article at hand; maybe it should move to a user talk page. — Carl (CBM · talk) 21:17, 29 October 2008 (UTC)
replies: text: Fine, I was merely alluding to the somewhat epistemological remark that the "real" nature of this famous line is somewhat of a misnomer, certainly according to modern scientific theory.  First of all, according to most modern accounts, the universe is finite, and therefore there is simply not enough room for R.  Even if you say that there is enough room for a big slice of it, quantum mechanics would come in to point out that the infinite divisibility that we assume in calculus simply does not agree with experiment, making it difficult (perhaps not impossible) to distinguish between Q and R, for example. Katzmik (talk) 10:05, 30 October 2008 (UTC)
thread_title:  reality versus hyperreality 

message: We currently have two alternative characterisations, one in terms of non-standard analysis, and one in terms of pairs of sequences.  It may be helpful if someone with expertise in this area could write a sentence pointing out the equivalence of these two approaches, as sequences, together with an ultrafilter, are in fact used to define the hyperreals.  Katzmik (talk) 14:44, 30 October 2008 (UTC)
thread_title:  Alternative characterisations 

message: I don't understand why the template Calculus has been recently removed, there is no doubt that this topic is included in both advanced calculus and calculus courses and books: do we want to make a new template named Advanced calculus or a Topology one? --Jorgen W (talk) 11:40, 11 November 2008 (UTC)
thread_title:  Template  

message: As per recent edit by Thenub, I am not sure what vexes you concerning the alleged equivalence of the two definitions.  I explicitly wrote that the construction presented by Keisler is in terms of sequences.  Thus an infinitesimal can be thought of as represented by a sequence.  Easy examples appear in Keisler's Epilogue.  In this sense, there is a clear relation between the two definitions.  If certain words appear weasel, why not edit the sentence instead of removing it? Katzmik (talk) 16:10, 11 December 2008 (UTC)
replies: text: The construction outlined by Keisler also involves an equivalence relation that is not clearly the same as the condition on the sequences in the second definition.  They are not clearly equivalent, which is why I removed that statement that they may be equivalent.  Thenub314 (talk) 18:38, 11 December 2008 (UTC)
replies: text: You are right, "equivalent" was not the right word.  However, there is clearly a relation here.  What do you think of a sentence of the following sort: "The two definitions below are related as both exploit sequences: the second definition explicitly, while the first, via the construction of hyperreals in terms of sequences as presented in the Epilogue".  This sounds too wordy, but it could be worked on. ?  Katzmik (talk) 20:28, 13 December 2008 (UTC)
thread_title:  equivalence of definitions 

message: We currently have two examples in this section, and maybe this should be part of the out of proportion discussion above, but here goes.  For the first one
:Example 1. The trigonometric tangent function, continuous on the open interval (−π/2, π/2) fails to be uniformly continuous on that set because there are infinitely small positive real numbers δ such that tan(π/2 − δ) and tan(π/2 − 2δ) differ by more than an infinitely small amount.
Is it intended to be clear to a reader why tan(π/2 − δ) and tan(π/2 − 2δ) differ by more than an infinitely small amount?
:Example 2. The exponential function exp, continuous on the real line R, fails to be uniformly continuous on R because there are infinitely large non-standard real numbers x and infinitely small non-standard real numbers δ such that exp(x + δ) − exp(x) is not infinitely small.  A more explicit calculation in the case of the squaring function x2 appears at Non-standard calculus.
Again why is it clear (from what is said) that these two infinite quantities are not infinitely close together?  I think that these two examples do not add much to the discussion of uniform continuity and should be re-worked or removed.  But maybe I should get other opinions first. Thenub314 (talk) 16:00, 12 December 2008 (UTC)
replies: text: Thanks.  My personal preference is for the quadratic example, is the calculation there is very explicit.  However, Michael is the editor who made this edit and he is more experienced in wiki issues than I am, perhaps we can wait to get some input from him.  Katzmik (talk) 20:30, 13 December 2008 (UTC)
thread_title:  Examples in Non-standard analysis characterization. 

message: I realize I'm 4 years late to the party, but I made two clarifying edits that I believe reconciles the two competing viewpoints of the fuzzy notion of a "local" property in the sections above. 173.227.48.5 (talk) 00:17, 14 December 2012 (UTC)
thread_title:  Local v Global, hyperreals 

message: The "specific" statement under Other characterisations/Cauchy continuity is the incorrect statement of Cauchy continuity -- in fact, the given mathematical statement is equivalent to uniform continuity. 
I made an  to restate the content of the section to state that the given form of uniform continuity perhaps makes the implication clearer, but was quickly undone.
Perhaps the section should be rewritten.
Chan-Paton factor (talk) 07:16, 25 December 2019 (UTC)
replies: text: Yes, I agree: the given mathematical statement is equivalent to uniform continuity (for arbitrary functions between arbitrary metric spaces). In contrast, Cauchy continuity is equivalent to uniform continuity on every totally bounded subset (for arbitrary functions between arbitrary metric spaces). If the first space is \mathbb R^n, then totally bounded subsets are just bounded subsets, and in this case Cauchy continuity is equivalent to uniform continuity on every bounded subset (which is sometimes called local uniform continuity). Boris Tsirelson (talk) 09:50, 25 December 2019 (UTC)
thread_title:  Incorrect characterisation of Cauchy continuity