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I've heard that Cauchy thought he'd proved that pointwise and uniform convergence are equivalent. Is this a historical fact? If it is indeed true, I was wondering if anyone had a reference.

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See answers or comments by me and Greg Graviton at… – KConrad Jul 21 '14 at 14:41
I have heard something along the lines of the following: Cauchy's conception of real numbers included infinitesimals. So for him pointwise convergence included infinitesimal pointwise convergence. In some rigorous sense, this does end up being uniform convergence. I believe this may be true in synthetic differential geometry, for instance. I probably read this in a MO comment somewhere, but I cannot find it. Maybe someone reading this could back up this perspective? – Steven Gubkin Jul 21 '14 at 18:02
@StevenGubkin, indeed uniform convergence is equivalent to a certain pointwise condition (in the extended domain) for the natural extensions of the functions. – Mikhail Katz Jun 16 at 14:14

See the wonderful book by Judith Grabiner, The Origins of Cauchy's Rigorous Calculus (1981), page 140 : "Actually, his [Cauchy's] proof implicitly assumed the function to be uniformly continuous, though he did not distinguish between continuity and uniform continuity, just as he had not distinguished between convergence and uniform convergence."

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Thanks for the reference to the book. In the link above, @GregGraviton argues that Cauchy knew the difference and this is because he includes infinitesimals in the definition of pointwise convergence. So perhaps it would be unfair to say, "Cauchy implicitly assumes uniform continuity". Instead, perhaps one should say, "infinitesimals are included in Cauchy's definition of pointwise convergence, and this can be shown to imply uniform convergence". Greg Graviton's argument appears compelling, and here is a paper about this. – arjun Jul 21 '14 at 18:39
@user128063: Feel free to edit my answer. I made it community wiki for this reason. – András Bátkai Jul 21 '14 at 19:27
@YemonChoi, I am basing myself on the quotation provided by Andras. The difference between uniform continuity and continuity is irrelevant here, as I explained. Furthermore you are misrepresenting my comment. I did not say that she "does not know the difference" but rathern that her comment here is irrelevant. It is mathematically incorrect to insist on the parallel between convergence and continuity since, unlike the case of convergence, there is no distinction between the two notions of continuity that's relevant to this case. – Mikhail Katz Jul 30 '14 at 10:15
@Yemon, thanks for your comment. I think you mean "syntactic distinction" rather than "semantic distinction" since semantically there is no difference between continuity and uniform continuity on a compact, at least in classical logic. But what's the meaning of your comment on "uniformity of the delta"? Grabiner explicitly acknowledges in the introduction to her book that Cauchy never gave an epsilon delta definition of continuity. He consistently gave an infinitesimal definition of continuity in all his publications... – Mikhail Katz Jul 30 '14 at 11:12
...Furthermore in 1853 Cauchy seems to extend his requirement to a larger domain by using the term "toujours", so if anything the relevant modern analogy would be between S-continuity at real points only versus at all points of the extended domain. – Mikhail Katz Jul 30 '14 at 11:13

The issue of Cauchy's understanding of continuity is a subject of lively historical debate. Grabiner represents only one view in this debate. Laugwitz has published a series of scholarly articles studying the issue. No discussion of this issue is complete without mentioning Cauchy's article dating from 1853 where he deals with convergence of series of functions, and seems to introduce a condition close to uniform convergence which is stronger than one used in his earlier works. See for instance this article.

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