Results 1 to 5 of 5

Thread: uniformly continuous

  1. #1
    Newbie
    Joined
    Oct 2016
    From
    india
    Posts
    13

    uniformly continuous

    Hello... i was trying to do this but stuck in between ...can anyone help me?

    F (x)= xsin (1/x): x not equal to 0
    F (x)= 0 : for x=0
    For all x belongs closed interval -1,1..
    Prove f (x) is uniformly continuous by using defination?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,360
    Thanks
    2680
    Awards
    1

    Re: uniformly continuous

    Quote Originally Posted by hanish View Post
    Hello... i was trying to do this but stuck in between ...can anyone help me?

    F (x)= xsin (1/x): x not equal to 0
    F (x)= 0 : for x=0
    For all x belongs closed interval -1,1..
    Prove f (x) is uniformly continuous by using defination?
    After many years of lecturing on this material, I don't remember seeing a $\epsilon/\delta$ proof of this. What I have seen many times is the fact that $F(x)$ defined above is continuous on $[0,1]$ therefore it is uniformly continuous. If you find a direct proof, then I would like to see it.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Oct 2016
    From
    india
    Posts
    13

    Re: uniformly continuous

    Dear plato..i have seen its solution in maths forum but i am not satisfied with that..
    There in defination they had taken
    |x1Sin (1/x1)-x2sin (1/x2)| is less than or equal to |x1sin (1/x1)-x2sin (1/x1)+x1sin (1/x2)-x2sin (1/x2)|
    And then simplifying RHS we get 2 |x1-x2|...
    They had proved in this way but i was thinking how middle both term is always greater than or equal to LHS .. N the reason they had given was that
    Both extra added term are almost equal..how they can be equal?? Can you please help me?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,336
    Thanks
    2857

    Re: uniformly continuous

    Quote Originally Posted by hanish View Post
    Dear plato..i have seen its solution in maths forum but i am not satisfied with that..
    There in € defination they had taken
    |x1Sin (1/x1)-x2sin (1/x2)| is less than or equal to |x1sin (1/x1)-x2sin (1/x1)+x1sin (1/x2)-x2sin (1/x2)|
    That's not correct- x2sin(1/x1) and x1 sin(1/x2) will not cancel. That last term should be |x1sin(1/x1)- x2sin(1/x1)+ x2sin(1/x1)- x2sin(1/x2)|= |(x1- x2)sin(1/x1)+ x2(sin(1/x1)- sin(1/x2))|.

    And then simplifying RHS we get 2 |x1-x2|...
    They had proved in this way but i was thinking how middle both term is always greater than or equal to LHS .. N the reason they had given was that
    Both extra added term are almost equal..how they can be equal?? Can you please help me?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Oct 2016
    From
    india
    Posts
    13

    Re: uniformly continuous

    Dear hallsofIvy by doing this we are getting desired answer or not?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: Apr 18th 2011, 08:19 AM
  2. Replies: 3
    Last Post: Apr 18th 2011, 07:24 AM
  3. Uniformly Continuous but not absolutely continuous example?
    Posted in the Advanced Math Topics Forum
    Replies: 1
    Last Post: Mar 9th 2010, 01:23 PM
  4. Uniformly Continuous but not absolutely continuous example?
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Mar 9th 2010, 10:43 AM
  5. Uniformly continuous?
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Oct 25th 2009, 07:18 PM

/mathhelpforum @mathhelpforum