Results 1 to 3 of 3

Math Help - example of function

  1. #1
    Junior Member
    Joined
    Mar 2009
    Posts
    44

    example of function

    Can anybody give me an example of function which is holder continuous but is not bounded variation function?
    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by sidi View Post
    Can anybody give me an example of function which is holder continuous but is not bounded variation function?
    I think that the function x\sin\frac1x will satisfy both conditions. It is not of bounded variation, because it varies by an amount of approximately 1/k in each interval of the form \Bigl[\frac1{(2k+\frac12)\pi},\frac1{(2k-\frac12)\pi}\Bigr]. And it ought to satisfy a Hölder continuity condition with exponent 1/2 (or maybe a bit less).

    I haven't checked carefully that the above example works. For an alternative construction of an example which definitely does work, look at q.2 in this pdf file.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    Quote Originally Posted by Opalg View Post
    For an alternative construction of an example which definitely does work, look at q.2 in this pdf file.
    I have a problem with this example, I don't really understand what the function in the pdf is supposed to be like. But more importantly I have this:

    Since W^{1,p}(0,1)=C^{0,\alpha}(0,1) as sets (obviously we pick just one representative in the class of u\in W^{1,p}(0,1)) for p> 1, \alpha = 1-\frac{1}{p} and because W^{1,p}(0,1)\subset W^{1,1}(0,1)=AC(0,1)\subset BV(0,1) for all p\geq 1 (the only thing we need here is the interval to be bounded) so, since we might as well have picked [0,1] instead, we would get that every Hölder continous function over a bounded interval is of bounded variation, which would contradict your examples.

    I'm not sure I haven't made a mistake, but not really understanding the pdf example I can't say for sure where I made a mistake (or where he did).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 20
    Last Post: November 27th 2012, 06:28 AM
  2. Replies: 3
    Last Post: November 29th 2011, 03:08 PM
  3. Replies: 0
    Last Post: October 19th 2011, 05:49 AM
  4. Replies: 4
    Last Post: October 27th 2010, 06:41 AM
  5. Replies: 3
    Last Post: September 14th 2010, 03:46 PM

Search Tags


/mathhelpforum @mathhelpforum