Results 1 to 3 of 3
Like Tree2Thanks
  • 1 Post By Gusbob
  • 1 Post By Plato

Math Help - Real Analysis Prove discontinuous

  1. #1
    Newbie
    Joined
    Mar 2013
    From
    sacramento
    Posts
    7

    Real Analysis Prove discontinuous

    f: [0,1]--> R (reals)

    f(x) = {1 if x= 1/n where nEN (element Naturals) , 0 otherwise}

    Prove f is discontinuous at 0

    I have not been able to make any progress on this problem and need help please.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Jan 2008
    Posts
    588
    Thanks
    87

    Re: Real Analysis Prove discontinuous

    Quote Originally Posted by leonhart View Post
    f: [0,1]--> R (reals)

    f(x) = {1 if x= 1/n where nEN (element Naturals) , 0 otherwise}

    Prove f is discontinuous at 0

    I have not been able to make any progress on this problem and need help please.
    \displaystyle{\lim_{n\to \infty}}f\left(\frac{1}{n}\right)= 1 \to 1 \not= f(0)=0
    Thanks from leonhart
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,614
    Thanks
    1581
    Awards
    1

    Re: Real Analysis Prove discontinuous

    Quote Originally Posted by leonhart View Post
    f: [0,1]--> R (reals)
    f(x) = {1 if x= 1/n where nEN (element Naturals) , 0 otherwise}
    Prove f is discontinuous at 0
    A word about notation. This problem assumes that \mathbb{N}=\mathbb{Z}^+, that is not always the case.

    The function is f(x)=\begin{cases}\dfrac{1}{x} &:\; x\in\mathbb{Z}^+\\0 &:\; \text{otherwise}\end{cases}.

    Note that {\lim _{x \to {0^ + }}}f(x) = \infty . Therefore f cannot be continuous at x=0.

    If you need a \varepsilon;\delta proof note that \left( {\forall \varepsilon  > 0} \right)\left( {\exists J \in {\mathbb{Z}^ + }} \right)\left[ {0 < {J^{ - 1}} < \varepsilon } \right]
    Thanks from leonhart
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Prove This Is Discontinuous At Zero
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: May 24th 2010, 05:01 PM
  2. real analysis
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 7th 2010, 06:30 AM
  3. Real Analysis: Prove BV function bounded and integrable
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: January 17th 2010, 02:31 PM
  4. real analysis...pls help me prove this!
    Posted in the Calculus Forum
    Replies: 3
    Last Post: August 25th 2008, 09:37 AM
  5. Replies: 4
    Last Post: October 3rd 2007, 05:28 PM

Search Tags


/mathhelpforum @mathhelpforum