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Henry1
science forum addict
Joined: 15 May 2005
Posts: 58
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 Posted: Thu May 26, 2005 10:25 pm Post subject:
Re: strictly decreasing function with zero derivative on dense subset
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On Thu, 26 May 2005 06:22:46 -0500, David C. Ullrich
<ullrich@math.okstate.edu> wrote:
| Quote: | On Wed, 25 May 2005 23:16:48 +0000 (UTC), Henry <se16@btinternet.com
wrote:
Take the interval [0,1]. Then exclude all points of the form a/2^m
(with a and m positive integers and a<2^m) and the intervals around
them [a/2^m - k/4^m, a/2^m + k/4^m] for some real k with 0<=k<=2.
For 0<k<2 this leaves a nowhere dense set with positive measure.
Let f(k) be the measure of that set - this has the desired properties:
in particular f'(k)=0 when k is of the form b/2^(n-1) with b and n
positive integers and b<2^n.
Huh. Two questions:
(i) Is it obvious that f'(k) = 0 for such k? (It's not immediately
clear to me, but I haven't thought about it enough to expect it
would be clear.)
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The reason is that for such k the end point of each removed interval
falls in the middle of a smaller removed interval so small changes in
k lead to much smaller changes in the measure.
| Quote: | (ii) Is this function differentiable _everywhere_? (Not that it's
relevant to the original question in this thread, but the point
to the example in Stromberg is that the function _is_ differentiable
at every point - if we settle for f almost everywhere differentiable
then it's much easier to get f' = 0 on a dense set.)
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No: it is left- and right-differentiable everywhere, but these are
unequal at a countable set of points where the end point of a removed
interval coincides with the end point of a smaller removed interval.
Simple examples are when k=4/3 or k=4/5. |
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David C. Ullrich
science forum Guru
Joined: 28 Apr 2005
Posts: 2250
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| Posted: Thu May 26, 2005 9:22 am Post subject:
Re: strictly decreasing function with zero derivative on dense subset
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On Wed, 25 May 2005 23:16:48 +0000 (UTC), Henry <se16@btinternet.com>
wrote:
| Quote: | On Wed, 25 May 2005 09:57:53 GMT, Kira Yamato <no@mail.net> wrote:
José Carlos Santos wrote:
On 25-05-2005 2:32, Henry wrote:
I seem to have a continuous strictly decreasing function
f(x):[0,2]->[0,1] which seems to have a derivative f'(x)=0 for x in a
dense subset of [0,2]. It that possible?
There's an example of such a funcion in Stromberg's "Introduction to
Classical Real Analysis", for instance. Well, actually what he gives is
an example of a strictly *increasing* function whose derivative is 0 in
a dense set, but... :-)
I think the example you mentioned is this one: let {r_n} be the
rationals. Define
f(x) := x + sum_n (x-r_n)^(1/3)/(n^2 (1+ |r_n|)^(1/3)).
Then the inverse function F of f has that property.
Mine is rather different:
Take the interval [0,1]. Then exclude all points of the form a/2^m
(with a and m positive integers and a<2^m) and the intervals around
them [a/2^m - k/4^m, a/2^m + k/4^m] for some real k with 0<=k<=2.
For 0<k<2 this leaves a nowhere dense set with positive measure.
Let f(k) be the measure of that set - this has the desired properties:
in particular f'(k)=0 when k is of the form b/2^(n-1) with b and n
positive integers and b<2^n.
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Huh. Two questions:
(i) Is it obvious that f'(k) = 0 for such k? (It's not immediately
clear to me, but I haven't thought about it enough to expect it
would be clear.)
(ii) Is this function differentiable _everywhere_? (Not that it's
relevant to the original question in this thread, but the point
to the example in Stromberg is that the function _is_ differentiable
at every point - if we settle for f almost everywhere differentiable
then it's much easier to get f' = 0 on a dense set.)
************************
David C. Ullrich |
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Henry1
science forum addict
Joined: 15 May 2005
Posts: 58
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| Posted: Wed May 25, 2005 11:16 pm Post subject:
Re: strictly decreasing function with zero derivative on dense subset
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On Wed, 25 May 2005 09:57:53 GMT, Kira Yamato <no@mail.net> wrote:
| Quote: | José Carlos Santos wrote:
On 25-05-2005 2:32, Henry wrote:
I seem to have a continuous strictly decreasing function
f(x):[0,2]->[0,1] which seems to have a derivative f'(x)=0 for x in a
dense subset of [0,2]. It that possible?
There's an example of such a funcion in Stromberg's "Introduction to
Classical Real Analysis", for instance. Well, actually what he gives is
an example of a strictly *increasing* function whose derivative is 0 in
a dense set, but... :-)
I think the example you mentioned is this one: let {r_n} be the
rationals. Define
f(x) := x + sum_n (x-r_n)^(1/3)/(n^2 (1+ |r_n|)^(1/3)).
Then the inverse function F of f has that property.
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Mine is rather different:
Take the interval [0,1]. Then exclude all points of the form a/2^m
(with a and m positive integers and a<2^m) and the intervals around
them [a/2^m - k/4^m, a/2^m + k/4^m] for some real k with 0<=k<=2.
For 0<k<2 this leaves a nowhere dense set with positive measure.
Let f(k) be the measure of that set - this has the desired properties:
in particular f'(k)=0 when k is of the form b/2^(n-1) with b and n
positive integers and b<2^n. |
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Kira Yamato
science forum beginner
Joined: 03 May 2005
Posts: 37
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| Posted: Wed May 25, 2005 7:57 am Post subject:
Re: strictly decreasing function with zero derivative on dense subset
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José Carlos Santos wrote:
| Quote: | On 25-05-2005 2:32, Henry wrote:
I seem to have a continuous strictly decreasing function
f(x):[0,2]->[0,1] which seems to have a derivative f'(x)=0 for x in a
dense subset of [0,2]. It that possible?
There's an example of such a funcion in Stromberg's "Introduction to
Classical Real Analysis", for instance. Well, actually what he gives is
an example of a strictly *increasing* function whose derivative is 0 in
a dense set, but... 
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I think the example you mentioned is this one: let {r_n} be the
rationals. Define
f(x) := x + sum_n (x-r_n)^(1/3)/(n^2 (1+ |r_n|)^(1/3)).
Then the inverse function F of f has that property.
Really cool stuff.
-kira
| Quote: |
Best regards,
Jose Carlos Santos |
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José Carlos Santos
science forum Guru
Joined: 25 Mar 2005
Posts: 1111
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| Posted: Wed May 25, 2005 6:03 am Post subject:
Re: strictly decreasing function with zero derivative on dense subset
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On 25-05-2005 2:32, Henry wrote:
| Quote: | I seem to have a continuous strictly decreasing function
f(x):[0,2]->[0,1] which seems to have a derivative f'(x)=0 for x in a
dense subset of [0,2]. It that possible?
|
There's an example of such a funcion in Stromberg's "Introduction to
Classical Real Analysis", for instance. Well, actually what he gives is
an example of a strictly *increasing* function whose derivative is 0 in
a dense set, but... :-)
Best regards,
Jose Carlos Santos |
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Justin Young
science forum addict
Joined: 02 May 2005
Posts: 55
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| Posted: Wed May 25, 2005 1:32 am Post subject:
Re: strictly decreasing function with zero derivative on dense subset
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"Henry" <se16@btinternet.com> wrote in message
news:02l791911o30innftv84rb6pqc6ei6bjih@4ax.com...
| Quote: | I seem to have a continuous strictly decreasing function
f(x):[0,2]->[0,1] which seems to have a derivative f'(x)=0 for x in a
dense subset of [0,2]. It that possible?
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Yes. |
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Henry1
science forum addict
Joined: 15 May 2005
Posts: 58
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| Posted: Wed May 25, 2005 1:32 am Post subject:
strictly decreasing function with zero derivative on dense subset
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I seem to have a continuous strictly decreasing function
f(x):[0,2]->[0,1] which seems to have a derivative f'(x)=0 for x in a
dense subset of [0,2]. It that possible? |
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