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James1118
science forum Guru Wannabe

Joined: 04 Feb 2005
Posts: 154

Posted: Tue Jul 11, 2006 4:54 pm    Post subject: Absolutely continuous, xsin(1/x)

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

James
Dave L. Renfro
science forum Guru

Joined: 29 Apr 2005
Posts: 570

Posted: Tue Jul 11, 2006 5:09 pm    Post subject: Re: Absolutely continuous, xsin(1/x)

James wrote:

 Quote: 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.)

in a first semester calculus course. (Look up the
squeeze theorem, also called the sandwich theorem.)

Dave L. Renfro
amzoti
science forum beginner

Joined: 18 Apr 2006
Posts: 33

Posted: Tue Jul 11, 2006 5:11 pm    Post subject: Re: Absolutely continuous, xsin(1/x)

James wrote:
 Quote: 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
science forum Guru

Joined: 28 Apr 2005
Posts: 2250

Posted: Tue Jul 11, 2006 8:56 pm    Post subject: Re: Absolutely continuous, xsin(1/x)

On 11 Jul 2006 10:11:59 -0700, "amzoti" <amzoti@yahoo.com> wrote:

 Quote: 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.

 Quote: 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
science forum Guru Wannabe

Joined: 06 May 2005
Posts: 245

Posted: Tue Jul 11, 2006 9:24 pm    Post subject: Re: Absolutely continuous, xsin(1/x)

James wrote:
 Quote: 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
science forum Guru Wannabe

Joined: 06 May 2005
Posts: 245

Posted: Tue Jul 11, 2006 9:30 pm    Post subject: Re: Absolutely continuous, xsin(1/x)

Jiri Lebl wrote:
 Quote: 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?

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
Frreita

Joined: 13 Jun 2005
Posts: 96

Posted: Tue Jul 11, 2006 10:14 pm    Post subject: Re: Absolutely continuous, xsin(1/x)

"James" <james545@gmail.com> wrote in message
news:1251361.1152636874444.JavaMail.jakarta@nitrogen.mathforum.org...
 Quote: 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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