Possible proof of Gabriel's Theorem?
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From: Angus Rodgers on 12 Mar 2005 08:31 On Fri, 11 Mar 2005 12:55:08 +0000, I wrote: >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 .... again ... >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. I should mention that this is the Dover edition, which is a corrected reprint (2003) of the Holden-Day (1964) book with the title /Counterexamples in Analysis/. Elsewhere, "John Gabbriel" has referred to a Springer (1990, corr. repr. 1993) edition of a (presumably much expanded and revised) book with a new title /Theorems and Counterexamples in Mathematics/. (I hadn't heard of this book before, which is why I confused the vaguely recollected reference with the older book which I happened to have ordered recently.) -- Angus Rodgers (angus_prune@ eats spam; reply to angusrod@) Contains mild peril
From: Zbigniew Fiedorowicz on 12 Mar 2005 11:17 Angus Rodgers wrote: > On Fri, 11 Mar 2005 11:38:09 +0000, I wrote: > 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?): Given a nonconstant everywhere differentiable function f(x) on the unit interval with f'(x)=0 on a dense countable subset D, it is not difficult to modify it to one whose derivative vanishes on all the rationals. First construct a sequence of C^1 functions g_n(x) from the unit interval to itself having the following properties: (1) g_n(x) uniformly converge to a function g(x) (2) g'_n(x) uniformly converge to a function h(x) (3) For any fixed rational number r in the unit interval, the sequence g_n(r) becomes eventually constant with value in D. Then g(x) is easily seen to be C^1 with derivative g'(x)=h(x). Then the composite function fg is a nonconstant everywhere differentiable function on the unit interval, whose derivative vanishes on all the rational numbers. [Note that I am not claiming that (fg)' vanishes only on the rationals.] > 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. > > [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.) It (actually a different edition) is available online in the University of Michigan Historical Math Collection at the following url: http://www.hti.umich.edu/t/text/gifcvtdir/acm2112.0001.001/00000641.tif.20.pdf (This url points to the relevant pages: 626-634.)
From: Jason on 12 Mar 2005 11:18 > 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. I should mention that this is the Dover edition, which is a > corrected reprint (2003) of the Holden-Day (1964) book with the title /Counterexamples in Analysis/ As I thought... > Elsewhere, "John Gabbriel" has referred to a Springer (1990, corr. repr. 1993) edition of a (presumably much expanded and > revised) book with a new title /Theorems and Counterexamples in Mathematics/. (I hadn't heard of this book before, which > is why I confused the vaguely recollected reference with the older book which I happened to have ordered recently.) I would take whatever this fellow "John Gabbriel" says with a pinch of salt. Jason Wells.
From: David C. Ullrich on 12 Mar 2005 14:08 On Sat, 12 Mar 2005 11:17:07 -0500, Zbigniew Fiedorowicz <fiedorow(a)hotmail.com> wrote: >Angus Rodgers wrote: > >> On Fri, 11 Mar 2005 11:38:09 +0000, I wrote: >> 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?): > >Given a nonconstant everywhere differentiable function f(x) on the unit >interval with f'(x)=0 on a dense countable subset D, it is not difficult >to modify it to one whose derivative vanishes on all the rationals. > >First construct a sequence of C^1 functions g_n(x) from the unit >interval to itself having the following properties: >(1) g_n(x) uniformly converge to a function g(x) (1.5) And g is non-constant. >(2) g'_n(x) uniformly converge to a function h(x) >(3) For any fixed rational number r in the unit interval, the > sequence g_n(r) becomes eventually constant with value in D. >Then g(x) is easily seen to be C^1 with derivative g'(x)=h(x). > >Then the composite function fg is a nonconstant everywhere >differentiable function on the unit interval, whose derivative vanishes >on all the rational numbers. [Note that I am not claiming that (fg)' >vanishes only on the rationals.] Well that was pretty simple. You're also not claiming that g maps the rationals _onto_ D, which is what I was trying to do for some reason... >> 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. >> >> [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.) > >It (actually a different edition) is available online >in the University of Michigan Historical Math >Collection at the following url: >http://www.hti.umich.edu/t/text/gifcvtdir/acm2112.0001.001/00000641.tif.20.pdf >(This url points to the relevant pages: 626-634.) > ************************ David C. Ullrich
From: Jason on 12 Mar 2005 14:42
Hate to tell you this: This example would apply to the ftoc as well. So what bullshit are you trying to push here?! I can also come up with a number of functioins that don't normally work by producing some of my own mapings. Ullrich! You are so damn pathetic, it's just not funny! Is this your counter-example? Gee, no wonder your students are all idiots!! Jason Wells |