Possible proof of Gabriel's Theorem?
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From: Angus Rodgers on 13 Mar 2005 08:57 On Sun, 13 Mar 2005 06:22:22 -0600, David C. Ullrich <ullrich(a)math.okstate.edu> wrote: >On 12 Mar 2005 11:42:17 -0800, "Jason" <logamath(a)yahoo.com> wrote: > >>[****] > >You can find examples in various books, for example Stromberg >"Introduction to Classical Real Analysis", of strictly increasing >differentiable f with f' = 0 on a dense set. Is Stromberg's example simpler than the huge one given by Hobson? (I must admit I'm not yet willing to take the time needed to read through all of that one, although it looks like fun.) >>Gee, no wonder your students are all idiots!! Duh, did he say something bad, Boss? >You say things like this and worse really a lot. Do you keep >that in mind when you complain about the personal attacks that >you imagine you're getting here? I'll attack him again, if you want. Just say da word, Boss. Da punk ain't got no respect. -- Angus Rodgers (angus_prune@ eats spam; reply to angusrod@) Contains mild peril
From: David C. Ullrich on 13 Mar 2005 10:20 On Sun, 13 Mar 2005 13:57:00 +0000, Angus Rodgers <angus_prune(a)bigfoot.com> wrote: >On Sun, 13 Mar 2005 06:22:22 -0600, David C. Ullrich ><ullrich(a)math.okstate.edu> wrote: > >>On 12 Mar 2005 11:42:17 -0800, "Jason" <logamath(a)yahoo.com> wrote: >> >>>[****] >> >>You can find examples in various books, for example Stromberg >>"Introduction to Classical Real Analysis", of strictly increasing >>differentiable f with f' = 0 on a dense set. > >Is Stromberg's example simpler than the huge one given by Hobson? I don't know, but I suspect so. Hobson is the example that does this and much more, right? The example in Stromberg is an exercise that takes a half page or so. >(I must admit I'm not yet willing to take the time needed to read >through all of that one, although it looks like fun.) > >>>Gee, no wonder your students are all idiots!! > >Duh, did he say something bad, Boss? > >>You say things like this and worse really a lot. Do you keep >>that in mind when you complain about the personal attacks that >>you imagine you're getting here? > >I'll attack him again, if you want. Just say da word, Boss. >Da punk ain't got no respect. ************************ David C. Ullrich
From: Angus Rodgers on 13 Mar 2005 11:18 On Sun, 13 Mar 2005 09:20:32 -0600, David C. Ullrich <ullrich(a)math.okstate.edu> wrote: >On Sun, 13 Mar 2005 13:57:00 +0000, Angus Rodgers ><angus_prune(a)bigfoot.com> wrote: > >>On Sun, 13 Mar 2005 06:22:22 -0600, David C. Ullrich >><ullrich(a)math.okstate.edu> wrote: >> >>>On 12 Mar 2005 11:42:17 -0800, "Jason" <logamath(a)yahoo.com> wrote: >>> >>>>[****] >>> >>>You can find examples in various books, for example Stromberg >>>"Introduction to Classical Real Analysis", of strictly increasing >>>differentiable f with f' = 0 on a dense set. >> >>Is Stromberg's example simpler than the huge one given by Hobson? > >I don't know, but I suspect so. Hobson is the example that >does this and much more, right? > >The example in Stromberg is an exercise that takes a half page >or so. Hobson (<http://www.hti.umich.edu/u/umhistmath/>, browse under `H', page 626): ``The first attempt to construct a function with maxima and minima in every interval, which should have at every point a finite differential coefficient, was made by Hankel. The function which he constructed is however not an everywhere-oscillating function. By Du Bois Reymond the view was expressed that no such function can exist, but Dini regarded the existence of such functions as highly probable. The first actual construction of such a function is due to Koepcke, who having first constructed an everywhere- oscillating function with derivatives on the right and on the left at every point, in a subsequent memoir obtained a function having the required properties. Koepcke's construction has been simplified by Pereno, and the account here given is based upon the work of the latter.'' The "simplified" construction occupies 9 pages of Hobson's treatise! Even so, it is not quite self-contained. The reference on page 629 to "section 398" is to a one-page proof of a general theorem on term-by-term differentiation. The reference on page 634 to "section 223" is to a summary of another 3 pages of (rather specialised-looking) results concerning everywhere-oscillating functions (and related topics). As I only skimmed, there may be other dependencies that I've missed. Life's too short! -- Angus Rodgers (angus_prune@ eats spam; reply to angusrod@) Contains mild peril
From: Zbigniew Fiedorowicz on 13 Mar 2005 15:11 Angus Rodgers wrote: > Is Stromberg's example simpler than the huge one given by Hobson? > > (I must admit I'm not yet willing to take the time needed to read > through all of that one, although it looks like fun.) Here is a fairly simple construction of an increasing everywhere differentiable function with a dense set of points on which the derivative vanishes. It is based on a post of Daniel Grubb (but with a confusing typo fixed). Let r_n be an enumeration of the rationals in [0,1]. Let f(x) be defined on the unit interval by f(x) = \sum_n=1^\infty (x-r_n)^{1/3}/2^n. Then f(x) is clearly continuous and strictly increasing. Consider the difference quotient Df(x,h) = [f(x+h)-f(x)]/h Then Df(x,h) = \sum_n=1^\infty phi_n(x,h)/2^n where phi_n(x,h)=1/(u_n^2 + u_n v_n + v_n^2), with u_n = (x-r_n)^{1/3} and v_n = (x+h-r_n)^{1/3}. We see that u_n^2 + u_n v_n + v_n^2 = (u_n/2 + v_n)^2 + 3u_n^2/4 From this it follows that 0 < phi_n(x,h) <= (4/3)(x-r_n)^{-2/3} Using this inequality, analyze the following three cases. Case 1: x is rational => Df(x,h)->\infty as h->0. Case 2: x is irrational and \sum_{n=1}^\infty (x-r_n)^{-2/3}/2^n diverges => Df(x,h)->\infty as h->0. Case 3: x is irrational and \sum_{n=1}^\infty (x-r_n)^{-2/3}/2^n converges => Df(x,h)-> \sum_{n=1}^\infty (1/3)(x-r_n)^{-2/3}/2^n>0 as h->0. Then the inverse function f^{-1}(x) is strictly increasing and everywhere differentiable on the interval f[0,1] and its derivative vanishes on the dense set f(rationals).
From: David C. Ullrich on 13 Mar 2005 18:22
On Sun, 13 Mar 2005 15:11:23 -0500, Zbigniew Fiedorowicz <fiedorow(a)hotmail.com> wrote: >Angus Rodgers wrote: > >> Is Stromberg's example simpler than the huge one given by Hobson? >> >> (I must admit I'm not yet willing to take the time needed to read >> through all of that one, although it looks like fun.) > >Here is a fairly simple construction of an increasing >everywhere differentiable function with a dense set of >points on which the derivative vanishes. It is based on >a post of Daniel Grubb (but with a confusing typo fixed). > >Let r_n be an enumeration of the rationals in [0,1]. >Let f(x) be defined on the unit interval by >f(x) = \sum_n=1^\infty (x-r_n)^{1/3}/2^n. > >Then f(x) is clearly continuous and strictly increasing. >Consider the difference quotient >Df(x,h) = [f(x+h)-f(x)]/h >Then Df(x,h) = \sum_n=1^\infty phi_n(x,h)/2^n >where phi_n(x,h)=1/(u_n^2 + u_n v_n + v_n^2), with >u_n = (x-r_n)^{1/3} and v_n = (x+h-r_n)^{1/3}. > >We see that >u_n^2 + u_n v_n + v_n^2 = (u_n/2 + v_n)^2 + 3u_n^2/4 > From this it follows that >0 < phi_n(x,h) <= (4/3)(x-r_n)^{-2/3} >Using this inequality, analyze the following three cases. > >Case 1: x is rational => Df(x,h)->\infty as h->0. > >Case 2: x is irrational and \sum_{n=1}^\infty (x-r_n)^{-2/3}/2^n >diverges => Df(x,h)->\infty as h->0. > >Case 3: x is irrational and \sum_{n=1}^\infty (x-r_n)^{-2/3}/2^n >converges => Df(x,h)-> \sum_{n=1}^\infty (1/3)(x-r_n)^{-2/3}/2^n>0 >as h->0. > >Then the inverse function f^{-1}(x) is strictly increasing and >everywhere differentiable on the interval f[0,1] and its derivative >vanishes on the dense set f(rationals). This _is_ more or less the example in Stromberg, with a few details worked out and/or changed. ************************ David C. Ullrich |