Possible proof of Gabriel's Theorem?
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From: Angus Rodgers on 11 Mar 2005 06:38 On Thu, 10 Mar 2005 08:25:19 -0600, David C. Ullrich <ullrich(a)math.okstate.edu> wrote: >On 9 Mar 2005 18:57:37 -0800, "Jason" <logamath(a)yahoo.com> wrote: > >>The mean value theorem has f and the integral has f'. >>Hint: Gabriel's theorem links the two! > >I don't know why you're still talking about Gabriel's >"theorem". It's simply _false_ - there have been at least >two counterexamples posted in the last day or so. Just checking (sorry if I'm being everywhere dense): I've seen WWWade's nice counterexample, in another thread, but have there been any other definite counterexamples? I've seen a reference to a book by Stromberg (of which even second-hand copies are very expensive), but I'm not sure if the application of this to Gabriel's non-theorem was worked out in detail. Also, "at least two"? Did someone mention a counterexample in Gelbaum & Olmsted, perhaps? (Amazon have just dispatched my copy, so I should be able to check for one there soon.) -- Angus Rodgers (angus_prune@ eats spam; reply to angusrod@) Contains mild peril
From: David C. Ullrich on 11 Mar 2005 07:24 On Fri, 11 Mar 2005 11:38:09 +0000, Angus Rodgers <angus_prune(a)bigfoot.com> wrote: >On Thu, 10 Mar 2005 08:25:19 -0600, David C. Ullrich ><ullrich(a)math.okstate.edu> wrote: > >>On 9 Mar 2005 18:57:37 -0800, "Jason" <logamath(a)yahoo.com> wrote: >> >>>The mean value theorem has f and the integral has f'. >>>Hint: Gabriel's theorem links the two! >> >>I don't know why you're still talking about Gabriel's >>"theorem". It's simply _false_ - there have been at least >>two counterexamples posted in the last day or so. > >Just checking (sorry if I'm being everywhere dense): I've >seen WWWade's nice counterexample, in another thread, but >have there been any other definite counterexamples? Yes, his example was in a reply to a post of mine where I gave a couterexample. >I've >seen a reference to a book by Stromberg (of which even >second-hand copies are very expensive), It's a very nice book. >but I'm not sure >if the application of this to Gabriel's non-theorem was >worked out in detail. No - I've looked at that, but I don't see how to adapt it to work here. In the book they get an increasing differentiable f with f' = 0 on a dense set, but I don't see how to get any control over _what_ dense set has f' = 0 using that construction, and here we need f' = 0 on the rationals to give the utterly obvious counterexample. >Also, "at least two"? Did someone >mention a counterexample in Gelbaum & Olmsted, perhaps? >(Amazon have just dispatched my copy, so I should be able >to check for one there soon.) ************************ David C. Ullrich
From: David C. Ullrich on 11 Mar 2005 07:26 On 10 Mar 2005 19:44:46 -0800, "Jason" <logamath(a)yahoo.com> wrote: >> How about going back and answering the questions. Stop >> ducking the question: > >How about you start answering my questions? For starters, show me where >gabriel's theorem has been stated before gabriel. Huh? Gabriel's theorem is _false_. Why would you expect to find the same error appearing previously? ************************ David C. Ullrich
From: Angus Rodgers on 11 Mar 2005 07:55 On Fri, 11 Mar 2005 11:38:09 +0000, I wrote: >Did someone >mention a counterexample in Gelbaum & Olmsted, perhaps? >(Amazon have just dispatched my copy, so I should be able >to check for one there soon.) Sorry to follow up my own post, but the book has just arrived. Unfortunately, unless I've missed something, it doesn't actually describe an example of a function with zero derivative at all rational points. However, there is a likely-looking reference on page 39 (could this even be to the same example as in Stromberg?): For an example of a function that is everywhere differentiable and nowhere monotonic, see [21], vol. II, pp. 412--421. Indeed, this last example gives a very elaborate construction of a function that is everywhere differentiable and has a dense set of relative maxima and a dense set of relative minima. [Footnote: See also A. Denjoy, /Bull. Soc. Math. France/, 43 (1915), pp. 161--248 (228ff.).] (France, 1915?! I'm imagining: "Hey, could you try to keep the noise down! I'm trying to concentrate on some maths here!") [21] Hobson, E. W., /The Theory of Functions/, Harren Press, Washington (1950). (I think this also used to be available as a Dover reprint, but is no longer.) -- Angus Rodgers (angus_prune@ eats spam; reply to angusrod@) Contains mild peril
From: Jason on 11 Mar 2005 10:30
Does this book contain a counter-example or does it contain *gabriel's theorem* ? Would it be too much trouble for you to scan this and post it on the web so we can look at it please? That is if it's not too many pages. Jason Wells |