FAQFAQ   SearchSearch   MemberlistMemberlist   UsergroupsUsergroups 
 ProfileProfile   PreferencesPreferences   Log in to check your private messagesLog in to check your private messages   Log inLog in 
Forum index » Science and Technology » Math
Absolutely continuous, xsin(1/x)
Post new topic   Reply to topic Page 1 of 1 [7 Posts] View previous topic :: View next topic
Author Message
James1118
science forum Guru Wannabe


Joined: 04 Feb 2005
Posts: 154

PostPosted: Tue Jul 11, 2006 4:54 pm    Post subject: Absolutely continuous, xsin(1/x) Reply with quote

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

James
Back to top
Dave L. Renfro
science forum Guru


Joined: 29 Apr 2005
Posts: 570

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

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.)

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
Back to top
amzoti
science forum beginner


Joined: 18 Apr 2006
Posts: 33

PostPosted: Tue Jul 11, 2006 5:11 pm    Post subject: Re: Absolutely continuous, xsin(1/x) Reply with quote

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.
Back to top
David C. Ullrich
science forum Guru


Joined: 28 Apr 2005
Posts: 2250

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

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
Back to top
Jiri Lebl
science forum Guru Wannabe


Joined: 06 May 2005
Posts: 245

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

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
Back to top
Jiri Lebl
science forum Guru Wannabe


Joined: 06 May 2005
Posts: 245

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

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
Back to top
Frreita
science forum addict


Joined: 13 Jun 2005
Posts: 96

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

"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 ?
Back to top
Google

Back to top
Display posts from previous:   
Post new topic   Reply to topic Page 1 of 1 [7 Posts] View previous topic :: View next topic
The time now is Thu Sep 21, 2017 11:16 pm | All times are GMT
Forum index » Science and Technology » Math
Jump to:  

Similar Topics
Topic Author Forum Replies Last Post
No new posts about uni. continuous proof bill1158 Math 1 Tue Jul 18, 2006 10:30 pm
No new posts Absolutely continuous, L^2 question James1118 Math 15 Wed Jul 12, 2006 1:00 pm
No new posts Properties of weakly continuous maps in a Hilbert space Jack Stone Math 10 Mon Jul 10, 2006 3:18 pm
No new posts Fixed points of a weakly continuous map f:H->H satisfying... Jack Stone Research 2 Mon Jul 10, 2006 3:12 pm
No new posts weakly continuous mapping in a Hilbert space f(f(x))=x Jack Stone Math 0 Sat Jul 08, 2006 3:04 pm

Copyright © 2004-2005 DeniX Solutions SRL
Other DeniX Solutions sites: Electronics forum |  Medicine forum |  Unix/Linux blog |  Unix/Linux documentation |  Unix/Linux forums  |  send newsletters
 


Powered by phpBB © 2001, 2005 phpBB Group
[ Time: 0.0847s ][ Queries: 16 (0.0666s) ][ GZIP on - Debug on ]