Results 1 to 3 of 3

Math Help - how to show if these are uniformly continuous

  1. #1
    Banned
    Joined
    Nov 2008
    Posts
    63

    how to show if these are uniformly continuous

     f(x) = xsinx, \ \ \ \ g(x)=e^{-x^{4}}
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by silversand View Post
     f(x) = xsinx, \ \ \ \ g(x)=e^{-x^{4}}

    The first one is not uniformly continous( I don't think)

    Here is number two

    Since g(x) is differentable g'(x)=-4x^3e^{-x^4} and

    the derivative is bounded on all of \mathbb{R}.

    so f'(x) < M for some M \in \mathbb{R}

    let \epsilon > 0 set \delta=\frac{\epsilon}{M}

    So now if |x-y|< \delta

    We need to show that |f(x)-f(y)|< \epsilon

    Now by the Mean Value theorem on [x,y]

    f(x)-f(y)=f'(c)(x-y)

    |f(x)-f(y)|=|f'(c)(x-y)|=|f'(c)||x-y|=f'(c)\cdot \delta =f'(c) \frac{\epsilon}{M}< \epsilon
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Dec 2008
    Posts
    130
    Quote Originally Posted by TheEmptySet View Post
    The first one is not uniformly continous( I don't think)

    Here is number two

    Since g(x) is differentable g'(x)=-4x^3e^{-x^4} and

    the derivative is bounded on all of \mathbb{R}.

    so f'(x) < M for some M \in \mathbb{R}

    let \epsilon > 0 set \delta=\frac{\epsilon}{M}

    So now if |x-y|< \delta

    We need to show that |f(x)-f(y)|< \epsilon

    Now by the Mean Value theorem on [x,y]

    f(x)-f(y)=f'(c)(x-y)

    |f(x)-f(y)|=|f'(c)(x-y)|=|f'(c)||x-y|=f'(c)\cdot \delta =f'(c) \frac{\epsilon}{M}< \epsilon
    the first one is not uniformly cont on \mathbb{R} b/c the derivative is xcos(x)+sin(x)(1). so f' \rightarrow \infty as x \rightarrow \infty contradicting the fact that the derivative must be bounded.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: April 18th 2011, 09:19 AM
  2. Replies: 3
    Last Post: April 18th 2011, 08:24 AM
  3. Show that sqrt(x) is uniformly continuous on [0,infinity)
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: March 25th 2010, 06:46 AM
  4. show this series converge uniformly to a continuous function
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 4th 2009, 01:07 AM
  5. Replies: 3
    Last Post: April 16th 2009, 04:09 PM

Search Tags


/mathhelpforum @mathhelpforum