Talk:Uniform continuity

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)

Continuous functions on a closed real interval

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)

You gave a wrong definition of uniformly continous

When the author writes

If $X$ and $Y$ are subsets of the real numbers, $d_{1}$ and $d_{2}$ can be the standard Euclidian norm, $|\cdot |$, yielding the definition: if there exists a $\delta >0$ such that, for all $\epsilon >0$ and $x,y\in X$, $|x-y|<\delta$ implies $|f(x)-f(y)|<\epsilon$.

s/he is trivially wrong. The right definition over the reals is

for every $\epsilon >0$, there exists a $\delta >0$ (depending on $\epsilon$!!!) such that, for all $x,y\in X$, $|x-y|<\delta$ implies $|f(x)-f(y)|<\epsilon$.

It's clearly a typo, since the author gave the right definition in the case of general metric spaces. If you use the definition I'm claiming to be wrong, even the identity function f(x)=x is not uniformly continous: let suppose that such a $\delta$ exists, and call it D. Let's take x and x+D/2, wich are closer than D. You will have |f(x)-f(x+D/2)| = D/2, wich is far to be smaller of any possible $\epsilon >0$... With the right definition, you will just take $\delta (\epsilon )=\epsilon$ and everything works fine. G2h2 (talk) 08:23, 7 April 2008 (UTC)

The comment is correct, the sentence

if there exists a $\delta >0$ such that, for all $\epsilon >0$ and $x,y\in X$, $|x-y|<\delta$ implies $|f(x)-f(y)|<\epsilon$.

should become

if for all $\epsilon >0$ there exists a $\delta >0$ such that, for all $x,y\in X$, $|x-y|<\delta$ implies $|f(x)-f(y)|<\epsilon$.

Stoppato (talk) 10:04, 7 April 2008 (UTC)

misleading lead?

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)

Some revisions/additions

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)

Let's discuss the place of nonstandard calculus in this article

I removed a sentence about uniform continuity in nonstandard calculus, for the second time. It seems only fair for me to discuss my reasons for doing so.

First, there is a mathematical point: it is claimed that the definition of uniform continuity in nonstandard calculus is "local." Exactly what this means is not explained, but to the extent that I understand the intended meaning I think that I disagree with it. A local property of a function f:X -> Y is a property that holds for the function iff for every point P in the domain, the property holds for the restriction of f to each element of an open neighborhood basis of that point (there are other, more elegant ways of phrasing this using things like "germs" and "stalks", but I hope the meaning is clear). In nonstandard calculus continuity is not (as I understand it) a special case of continuity in some topological space. Moreover, the definition of uniform continuity in nonstandard calculus still involves comparing two distinct points, as does the definition in standard calculus. In nonstandard calculus the points that one has to compare are "closer" than in standard calculus, but that doesn't change the logical form of the statement. This doesn't look like a local statement to me.

Second there is the encyclopedic point: we need to consider the policy WP:UNDUE of undue weight. In the lead on an article about uniform continuity, we should only be mentioning the most important core concepts related to uniform continuity. Subtle nuances, more advanced results, generalizations and so forth generally do not belong in the lead. I think the vast majority of mathematicians and students of mathematics at all levels (probably 99% or more) use the "standard" definition of uniform continuity, so the nonstandard version should not be referenced in the lead. On the other hand a section in the article on nonstandard calculus would be quite appropriate. Plclark (talk) 18:50, 26 October 2008 (UTC)

Please make an effort to understand the non-standard definition of uniform continuity before you make any further edits. The local nature of the definition of uniform continuity is clearly explained at uniform continuity. For instance, in the case of the interval (0,1], in addition to checking continuity at the standard points, one needs to check continuity at a positive infinitesimal. In other words, your assertion that you still need to deal with pairs of points rather than individual points, is incorrect.

The definition of nonstandard uniform continuity involves a condition that must be checked at pairs of points, just as the standard definition does. That it can be rephrased in terms of checking something at single "points" which are not points of the original space is unconvincing in regards to your claim that the property is local. By analogy, an increasing continuous function from [0,oo) is bounded iff it has a continuous extension to [0,oo]. To say that because of this the boundedness is a 'local' property on [0,oo) is problematic. Plclark (talk) 14:29, 27 October 2008 (UTC)

As far as your "vast majority" claim is concerned, I would agree that there is a standard majority. However, there is a significant minority that disagrees with your views.

To say 'significant minority' is, I think, quite misleading.

Wikipedia is not a dictatorship of the majority. I am willing to discuss this further, but please try to avoid further reverts based on insufficient understanding. Katzmik (talk) 08:18, 27 October 2008 (UTC)

Of course it isn't. Remember that research mathematicians do not accuse other research mathematicians of insufficient understanding. If there is something that I haven't understood well enough, then a sufficiently detailed content-related discussion will make this clear. I am not yet convinced that this is the case. Plclark (talk) 14:29, 27 October 2008 (UTC)

User Thenub followed the more elegant course of requesting a citation at (ε, δ)-definition of limit‎, which I duly provided. Katzmik (talk) 08:21, 27 October 2008 (UTC)

I agree with Plclark's points, and I have removed the contested sentence about non-standard calculus again. It should stay out of the article until and unless there is a consensus shown here for including it. Gandalf61 (talk) 09:36, 27 October 2008 (UTC)

It seems as though you also have misunderstood the mathematics, as you have numerous times on the talk page of non-standard calculus. Please refrain from further edits until you understand the material better. Katzmik (talk) 09:42, 27 October 2008 (UTC)

Katzmik has asked me to comment, so let it be noted that I agree that the nonstandard version should not be in the lede. Algebraist 12:55, 27 October 2008 (UTC)

My point is not necessarily that it should be mentioned in the lead, but that if a claim is made that the notion is "global", it should also be mentioned that this is a weakness of the standard Weierstrass approach, rather than an intrinsic feature of the property in question. Perhaps the entire local/global discussion should be postponed to a later section. Katzmik (talk) 13:03, 27 October 2008 (UTC)

The notion that the 'global' formulation of uniform continuity is a "weakness" seems decidedly POV to me: it would certainly need a citation. I don't understand the weakness: uniform continuity is simply not a local property in the sense that I mentioned above. What drawbacks does this have? Moreover, it is still not clear to me what precise definition of a local property is being used that makes nonstandard uniform continuity a local property. Plclark (talk) 14:29, 27 October 2008 (UTC)

Katzmik, I have no idea what you mean by saying that "a local definition of uniform continuity can be given in non-standard calculus". To me the statement sounds misleading or wrong, depending on what you mean. Let's consider the function $f(x)=x^{2}$. It seems to me that:

• f is not uniformly continuous.
• This is true also in the context of non-standard calculus.
• Every restriction of f to an interval of finite length is uniformly continuous.
• This is true also in the context of non-standard calculus.
• In the context of non-standard calculus, the restriction of f to the finite numbers is not uniformly continuous.
• In the context of non-standard calculus, the restriction of f to the finite numbers is an example of a function that is not uniformly continuous even though every element of the domain (in this case: every finite number) has an open neighbourhood on which the function is uniformly continuous.
• This shows that there is no local definition of uniform continuity.

The last item contradicts your statement. Could you please explain where the error in my reasoning is? Thanks. --Hans Adler (talk) 14:34, 27 October 2008 (UTC)

My thoughts:

• I looked over the content in question here and in Non-standard calculus, and I agree with Plcark's thoughts dated 14:29, 27 October 2008. I don't see that a local property of an extension of a function is a local property of the original function. So it may be true that in nonstandard analysis there is a local definition of uniform continuity of a total function, but that's a much weaker statement than saying that there is a local definition of uniform continuity of an arbitrary continuous function on some open subset of the space.
• Apart from books explicitly about nonstandard calculus, no real analysis text is going to discuss a potential local definition of uniform continuity via nonstandard analysis. It seems somewhat out of place to discuss it here, especially when it isn't clear there actually is a local definition of uniform continuity for the original function in nonstandard analysis.

Taking these points together, I don't think that the content under discussion should be included in this article. Of course it should be included in the article on nonstandard analysis. — Carl (CBM · talk) 17:03, 27 October 2008 (UTC)

Folks, please try to keep in mind that we are all ignoramuses to one extent or another when it comes to non-standard analysis (the two editors who have expertise in the matter don't seem interested in these nitty-gritties). For this reason, as I suggested above, it would be better actually to sort out the mathematics, before succmbing to the reverting instinct. The point is that the non-standard definition is a pointwise definition and therefore is more accessible than the standard definition which necessarily appeals to pairs of points. You may feel that the construction of the hyperreals is too heavy a price to pay for such improvements, but that's a separate point that could be discussed elsewhere. Katzmik (talk) 08:51, 28 October 2008 (UTC)

I have again removed a statement about the local definition of uniform continuity via non-standard calculus. One could make a similar definition in the standard setting. You could extend the domain of the function, by taking limits, to the support in the extend real numbers, and check continuity each and every point of the domain. This does not make the definition or the property local. I agree with many of the editors here that in either setting there is no local definition of uniform continuity. Which was why I asked for a citation about this subject in another article. Thenub314 (talk) 04:11, 28 October 2008 (UTC)

That's an interesting point. Would this make the identity function f(x)=x uniformly continuous? Apparently not. At any rate, if one can construct a standard pointwise definition of uniform continuity in some extended setting as you suggest, I would certainly withdraw the contention that non-standard analysis furnishes an advantage in this case. Katzmik (talk) 08:57, 28 October 2008 (UTC)

local versus global

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)

7-step argument

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)

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)

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)

Pointwise

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

Comment about nonstandard uniform continuity

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)

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)

I think it would be helpful for all of us to see a primary source, both to see more of the mathematics itself being presented and also to see in what way the material is presented. Accordingly I added a fact tag to the nonstandard characterization of uniform continuity.

Here is a mathematical comment: it seems that this is related to the issue of extension properties of uniformly continuous functions (I wrote this section in the current article, although I am not completely satisfied with the exposition). It seems to me that the "local" nonstandard criterion for uniform continuity on an interval like (0,1] amounts to saying that a function on this domain is uniformly continuous iff it has a continuous extension to 0 (which is, of course, true).

This certainly true :) and should be mentioned. Katzmik (talk) 10:36, 29 October 2008 (UTC)

(Again, whether one decides to call this a "pointwise" criterion is up for debate, but right now I'm just talking about mathematics, not how to write the article.) On the other hand, since a metric space like [0,oo) is already complete, uniform continuity is a weaker condition then extendability to the compactification [0,oo]: consider for instance $f(x)=x$. Here it seems that one needs to extend the range as well as the domain of the function, which is part of what is going on in the nonstandard version. Plclark (talk) 18:48, 28 October 2008 (UTC)

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)

an example

Here is a very trivial example that may be useful for comparison. Suppose f is continuous on [0,1). Extend this to a function from [0,1] to the extended real numbers by setting

$f(1)=\lim _{a\to 1}\sup _{x>a}|f(x)|$

Then the original function f is bounded on [0,1) if and only if the extension has a finite value at 1. Does this mean that if I permit myself to use extended real numbers then boundedness on an open interval is a local property of a continuous function? — Carl (CBM · talk) 12:39, 29 October 2008 (UTC)

I don't see anything wrong with saying so. Extending the domain works in the bounded case, so here the non-standard approach is not particularly useful. On the other hand, extending the domain does not work for R, whereas the non-standard definition can be formulated purely in terms of the pointwise germ of f*. Katzmik (talk) 12:42, 29 October 2008 (UTC)

The same procedure I just discussed would work on R. But I'm not comfortable saying that boundedness is a local property of a continuous function. After all, boundedness is one of the best examples I can think of a global, non-local property of a function... This particular extension of a continuous function to a new function involves surveying values of the original function at infinitely any points, so that examining the values of the extended function at the new points corresponds to examining the values of the original function at infinitely many points. But if I have to evaluate the extended function, it seems to me I am no longer talking about local properties of the original function, since the extension itself is a global property of the original function. — Carl (CBM · talk) 12:53, 29 October 2008 (UTC)

Sorry, I misread what you wrote. I thought you were still talking about uniform continuity. I am not sure why we are discussing boundedness, but I will make the following comment. When you define the boundary value on an extended domain, it is true that boundedness is then determined by the value at that point, but we are now dealing with a different function, whose definition is global. Katzmik (talk) 13:00, 29 October 2008 (UTC)

P.S. I would like to reproduce a comment I made elsewhere: you can also define a constant C to be 0 if f is uniformly continuous, and 1 if it is not. Then uniform continuity of f indeed becomes easily computable in terms of the value of C :) Katzmik (talk) 13:02, 29 October 2008 (UTC)

Yes, that's essentially what I'm doing. How is that different than the example of the extension of a function f to its nonstandard analogue f* ? Are you claiming that the value of f* on some infinite hyperreal is a local property of the original function f? That seems like a tough argument to me. I'm thinking of the potential argument that "using nonstandard analysis gives a local definition of uniform continuity for a real function f". — Carl (CBM · talk) 13:07, 29 October 2008 (UTC)

This is certainly a good point. I will have to think more about it, but let me make the following comment already. In more than one sense, when we are dealing with f*, we are dealing with the same function (I can elaborate on such senses). This is not so in the case of the objection you raised. Katzmik (talk) 13:14, 29 October 2008 (UTC)

In addition to this somewhat theoretical discussion, would you care to respond to the following query: take a look at the proof of the non-uniformity of x² at non-standard calculus, and let me know if you think a beginning student may not arrive at an intuition of what goes wrong with uniformity, at a lesser cost than reading the corresponding discussion in even the best-written standard calculus textbook? Katzmik (talk) 13:17, 29 October 2008 (UTC)

I would appreciate an expanded explanation on the precise way in which f and f* are related. This seems to be the main issue here. Regarding the x^2 example, I am too biased by my experience with standard calculus and my lack of experience with nonstandard analysis. I don't have any geometric image of a function on nonstandard hyperreals, and so I can't get a geometric image of uniform continuity from the definition that uses nonstandard methods. It looks to me that the main moral of the nonstandard definition is that it exploits the unboundedness of the derivative of x^2 so that at infinite hyperreals the "derivative" is actually infinite, allowing it to scale an infinitesimal to a non-infinitesimal. — Carl (CBM · talk) 13:37, 29 October 2008 (UTC)

As far as the issue of the exact relation between f and f*, the best solution would be to bring in some heavyweights at this point, I think. I should be able to provide adequate answers, but I think coming from the experts they may be more convincing. Do you have any history of fruitful interactions with CSTAR or Arthur Rubin? Katzmik (talk) 13:56, 29 October 2008 (UTC)

I have worked with them before, and they are both very professional. But I'm perfectly willing to accept answers from you, since I'll think through your comments the same as theirs. I do study logic, so I have quite a bit of relevant background.

You are ahead of me then. Katzmik (talk) 14:36, 29 October 2008 (UTC)

I'm just not intimately familiar with the particular ultrapower construction of the hyperreals. I'm familiar with the standard statement of Łoś's theorem but not with ultrapowers over typed languages, such as the language used to take arbitrary parameters from the superstructure V(R). But I don't think that level of technical detail is required for this particular discussion about global/local properties. — Carl (CBM · talk) 14:09, 29 October 2008 (UTC)

Concerning your derivative comment, there is nothing there about derivatives. Of course, we know this not to be the case cf. \sqrt{x} :) I am not sure how the derivative enters the picture. Add a bounded nowhere differentiable function to x^2, and exactly the same argument applies. Katzmik (talk) 13:59, 29 October 2008 (UTC)

The derivative only enters at the informal level of intuitive understanding. If you asked me for the "underlying reason" why x^2 is not uniformly continuous (in the standard setting), I would say it is because the derivative of x^2 goes to infinity as x goes to infinity. The nonstandard method described in the article seems to me to be a reflection of this fact, but instead of looking at a limit as x goes to infinity we are looking at a particular infinite hyperreal. — Carl (CBM · talk) 14:09, 29 October 2008 (UTC)

I would say that intuitively the point is that there points that are "very close" but at the same time the values of f are "not very close". Instead of formalizing this with quadruple quantifiers, for the past fifty years only we have been able to formalize it by saying that there are infinitely close points such that the values of f are not infinitely close, which seems closer to the intuitive statement than the standard formalisation. Katzmik (talk) 14:40, 29 October 2008 (UTC)

Surely it's the growth rate of how "not very close" the images can be, as different pairs of points at the same distance are considered? Also, I am skeptical that any intuition involving "infinitely close" points reflects the standard view, as 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. — Carl (CBM · talk) 14:45, 29 October 2008 (UTC)

Actually I have a lot of things to say about this, but I am trying to write a research paper simultaneously with this discussion. Would you mind discussing things one at a time? Let's focus on the f/f* issue for the time being if you don't mind. I certainly find it stimulating that people around wiki are getting more interested in Robinson's theory. Katzmik (talk) 14:48, 29 October 2008 (UTC) P.S. Perhaps this should be "Leibniz's theory"? :)

f and f*

A response to your question "I would appreciate an expanded explanation on the precise way in which f and f* are related" would seem to require a separate section :) Katzmik (talk) 14:42, 29 October 2008 (UTC)

I you don't mind I would like to start by copying some comments from the lead of transfer principle, whitten mostly by myself so all responsibility is mine:

In mathematics, the transfer principle is a concept in Abraham Robinson's non-standard analysis of the hyperreal numbers. It states that any sentence expressible in a certain formal language that is true of real numbers is also true of hyperreal numbers.

The transfer principle concerns the logical relation between the properties of the real numbers R, and the properties of a vastly larger field denoted *R called the hyperreals, constructed in terms of a standard axiomatisation of set theory such as ZFC. The field *R includes, in particular, entities that behave as infinitesimal ("infinitely small") numbers, providing a rigorous mathematical realisation of a project initiated by Leibniz.

The idea is to express virtually all of the analysis over R in a suitable language of mathematical logic, and then point out that this language applies equally well to *R. This turns out to be possible because at the set-theoretic level, the propositions in such a language are interpreted to apply only to internal sets rather than to all sets.

The theorem to the effect that each proposition valid over R, is also valid over *R, is called the transfer principle.

Let me know if this helps. Katzmik (talk) 14:44, 29 October 2008 (UTC)

According to the article nonstandard analysis the transfer principle only holds for bounded quantifier formulas, not for arbitrary formulas in the language. But that's a minor point. It is true that f* shares many definable properties with f, and that f* is uniquely obtained from f. But I don't see that this means that a local property of f* is also a local property of f, since the definition of f* at infinite nonstandard points must depend on all of f, rather than just depending on a bounded piece of f. — Carl (CBM · talk) 14:51, 29 October 2008 (UTC)

Just one comment regarding the above: Admittedly, the discussion at Internal set is insufficient. Hopefully you can improve it :) At any rate, Keisler's companion text "Foundations of infinitesimal calculus" has a section at the end where he discusses all this in detail. The text is over 200 pages but the foundational section is only about 20 or 30 pages, as I recall. Katzmik (talk) 14:52, 29 October 2008 (UTC)

I agree with your analysis of the definition of f*. Moreover, the construction is extremely global in a certain sense, since it is not sufficient to just build an ordered field with infinitesimals in it: we need a proof of the transfer principle for all the properties we would like to carry over. Incidentally, this is the reason I was never convinced by the surreal numbers: they don't seem to have a transfer principle, therefore their construction is merely a curiosity with no relation to, for example, the definition of the derivative, as there is no guarantee that any properties of y=f(x) will be carried over to dy/dx. At any rate, we are now talking about the construction of the hyperreals which is admittedly more difficult than anything like equivalence classes of Cauchy sequences, or Dedekind cuts. Yet it is a rare calculus class that deals in such niceties, and you always assume something "on faith". Even an honors calculus that does Dedekind cuts, will certainly not go over ZFC before defining derivatives. The fact that the construction of the hyperreals comes at a certain cost was one of the first things I wrote in reaction to your comments, as you recall. Katzmik (talk) 15:00, 29 October 2008 (UTC)

P.S. Similarly, the construction of the real using Dedekind cuts is extremely global: in fact, you need ALL the rationals to define a single real! I remember John Tate I think it was making the comment that if you think of a real number as an equivalence class of Cauchy sequences you will go bananas very quickly. This does not mean that the notion of doing things pointwise is meaningless over the reals. Similarly, when Leibniz thought about infinitesimals, he thought of individual points, even though the way they are constructed today is in terms of ultrafilters on the set of all sequences and the like. The non-standard definition of uniform continuity would most likely have been accepted by Leibniz as a pointwise one. Katzmik (talk) 16:41, 29 October 2008 (UTC)

I still can't see that a property that requires checking something at points not in the domain of a particular function would qualify as a "pointwise" property of that function. A pointwise property should only depend on what happens at points where the function is defined, not on what happens at points where the function is not defined. I think that my example about boundedness being a "local" property illustrates this.

Thanks for your comment. The hyperreal definition is certainly not a local definition in terms of f, but it is a local definition in terms of f*. I think the atmosphere is too charged right now to make any changes to the article, and at any rate I am perfectly happy with Hardy's edits. If we ever come back to this in the future, we can consider the possibility of adding a sentence to the effect that "in non-standard calculus, it is possible to give a local definition in terms of the natural extension f*". At any rate this is a minor point and Hardy's revision is excellent. He is also far more knowledgable about the subject matter than I am. Katzmik (talk) 10:01, 30 October 2008 (UTC)

As an aside, I don't think it's helpful to refer too much to Leibniz' views on infinitesimals, as I have never seen a convincing argument that modern nonstandard analysis actually does correspond to what Leibniz was doing. That is, I think it's somewhat revisionist to reinterpret Leibniz' work in terms of 20th century methods, just as it would be revisionist to interpret Euler's work on infinite series as if he had been familiar with the modern definitions of series convergence. — Carl (CBM · talk) 16:52, 29 October 2008 (UTC)

You are raising an interesting point. I would reply by saying that what we are discussing is not my opinion but rather Abraham Robinson's, as stated in his published work, namely Non-standard analysis, and quoted at Robinson's page (actually it is quoted at Non-standard analysis). Perhaps it is possible to argue that such a viewpoint is revisionist. Are you familiar with any published work that tries to argue this? I would certainly be interested in reading it. Katzmik (talk) 10:01, 30 October 2008 (UTC)

Out of proportion

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)

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)

Second, as far as I understand, generalising the non-standard definition to spaces more general than R and R* would not be straighforward (even possible?).

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)

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)

So the non-standard definition really covers only one special case (albeit important) of the concept of uniform continuity, which itself is not essentially more complicated in its general form than it is for real-valued functions on R.

Third, on the merits and locality of the definition for real-valued functions on R: while there is a certain sense in which the non-standard definition of continuity in terms of infinitesimal differences between finite elements in the domain mapping to infinitesimal differences between their images has certain appeal, I would not be so convinced of the pedagogical benefits of the non-standard definition of uniform continuity.

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)

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)

The reason is the need to appeal to infinitesimally close infinite hyperreals; I would argue that gaining actual insight about them is a few magnitudes more difficult than dealing with an additional quantifier in the definition. And these insights would appear to require resorting to the transfer principle. Indeed, it is instructive to follow the proof of the equivalence of the non-standard and the usual defition in Keisler's book mentioned above, with a simple example of continuous but not uniformly continuous function such as f(x) = x². To show that there are two infinitesimally close hyperreals x and y such that their images under f (or rather f*) are not infinitesimally close, one looks for pairs of finite ordinary reals a and b arbitrarily close to each other but such that their images are separated at least by a positive finite real number (say δ). Since these pairs can be found (by moving further and further away from zero, or "towards infinity"), one can invoke the transfer principle to claim that there is a pair consisting of infinitesimally close, necessarily infinite, hyperreals with the required property. Now it may be only me, but rather than finding the thus exhinited pair of infinite hyperrals helpful to understanding, I would need to decipher the geometric meaning of their existence, this time running through the highly non-trivial transfer principle.

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)

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)

All in all, I would leave the existence of a non-standard definition in this article as in interesting remark, wikilinking to the appropriate non-standard article, and leave it at that. Any discussion on the non-standard definition being "local" or "pointwise" would require first introducing (non-standard, I'm afraid) terminology in order to discuss an issue of only secondary importance to this article. Possibilities to elaborate on all this exist at the non-standard articles. Stca74 (talk) 20:14, 29 October 2008 (UTC)

reality versus hyperreality

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)

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)

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)

Alternative characterisations

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)

Template

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)

I removed it because 'uniform continuity' belongs to real analysis rather than calculus ('advanced calculus' is also 'real analysis').

Topology Expert (talk) 04:50, 12 November 2008 (UTC)

equivalence of definitions

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)

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)

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)

Examples in Non-standard analysis characterization.

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)

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)

Local v Global, hyperreals

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)

"Chain of Inclusions"

The article currently states, "Continuously differentiable ⊆ Lipschitz continuous ⊆ α-Hölder continuous ⊆ uniformly continuous ⊆ continuous." But f(x) = x² is a continuously differentiable function that is not uniformly continuous, right? Which would mean continuously differentiable does not imply uniformly continuous. 129.107.80.40 (talk) 23:38, 21 January 2016 (UTC)

I agree with this comment. Furthermore the article itself states that

"Every uniformly continuous function is continuous, but the converse does not hold."

GladiusMaximus34 (talk) 15:28, 30 March 2016 (UTC)

Did you mean that the article states "uniformly continuous = continuous"? Because that seems to be the problem you're describing, and I agree with you then. Deansg (talk) 06:23, 15 September 2017 (UTC)

Incorrect characterisation of Cauchy continuity

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 edit 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)

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)