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Absolutely continuous, xsin(1/x)

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James

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Jul 11, 2006, 12:54:04 PM7/11/06
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Is xsin(1/x) absolutely continuous on [0,1]? Is it continuous at 0?

James

Dave L. Renfro

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Jul 11, 2006, 1:09:56 PM7/11/06
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James wrote:

> Is xsin(1/x) absolutely continuous on [0,1]?
> Is it continuous at 0?

Can you show that x*sin(1/x) (defined to be zero at x=0)
fails to have bounded variation on [0,1]? (Knowing about
the divergence of the harmonic series will be useful.)

Your second question is one that is often addressed
in a first semester calculus course. (Look up the
squeeze theorem, also called the sandwich theorem.)

Dave L. Renfro

amzoti

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Jul 11, 2006, 1:11:59 PM7/11/06
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James wrote:
> Is xsin(1/x) absolutely continuous on [0,1]? Is it continuous at 0?
>
> James

Hello,

maybe I can make some suggestions.

1. Have you plotted the function over the specified range?

2. Have you looked at limits from 0- and 0+?

3. Have you looked at derivatives and concavity?

Basically, what have you tried and where are you stuck.

If this is HW, will providing an answer really serve your personal
growth?

Not sure that helps.

David C. Ullrich

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Jul 11, 2006, 4:56:32 PM7/11/06
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On 11 Jul 2006 10:11:59 -0700, "amzoti" <amz...@yahoo.com> wrote:

>
>James wrote:
>> Is xsin(1/x) absolutely continuous on [0,1]? Is it continuous at 0?
>>
>> James
>
>Hello,
>
>maybe I can make some suggestions.
>
>1. Have you plotted the function over the specified range?
>
>2. Have you looked at limits from 0- and 0+?
>
>3. Have you looked at derivatives and concavity?

_You_ can tell whether a function is absolutely continuous
by "plotting" it? Or by "looking at derivatives and
concavity"? I'd like to see how that works.

>Basically, what have you tried and where are you stuck.
>
>If this is HW, will providing an answer really serve your personal
>growth?
>
>Not sure that helps.


************************

David C. Ullrich

Jiri Lebl

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Jul 11, 2006, 5:24:03 PM7/11/06
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James wrote:
> Is xsin(1/x) absolutely continuous on [0,1]? Is it continuous at 0?
>
> James

Hints: Try looking at all the places where sin(1/x) is 1 or -1. For
example
sin(2pi*n + (1/2)pi) = 1 for all n. Can you estimate the variation of
this function from below by using this fact?

Jiri

Jiri Lebl

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Jul 11, 2006, 5:30:18 PM7/11/06
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Of course the hint is for the first question only. For the second
question just notice that for
x != 0 we have |x sin (1/x)| <= |x| since sin(y) <= 1 for all y. Now
apply the delta epsion definition of continuity. To estimate sines and
cosines by 1 is a fairly common trick that will help you with many
problems from basic analysis.

Jiri

Norm Grimpo

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Jul 11, 2006, 6:14:07 PM7/11/06
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"James" <jame...@gmail.com> wrote in message
news:1251361.11526368744...@nitrogen.mathforum.org...

> Is xsin(1/x) absolutely continuous on [0,1]? Is it continuous at 0?
>
> James

which from which slope/direction does the function approach 0 ?


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