Results 1 to 5 of 5

Math Help - Non continuous function

  1. #1
    Junior Member
    Joined
    Oct 2012
    From
    Israel
    Posts
    26

    Non continuous function

    Hey everyone,
    How would you prove that this function is not continuous in every x != 1:
    x-1 +((x-1)^2)*D(x)

    D(x) is Dirichlet function.
    if x rational -> D(x) = 0
    if not -> D(x) = 1

    Thanks a lot,
    Michael Engslter
    Last edited by MichaelEngstler; November 9th 2012 at 06:09 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,427
    Thanks
    1857

    Re: Non continuous function

    There are several different "Dirichlet functions" but I assume you mean the most common, D(x)= 1 if x is rational, 0 otherwise. Given any x, there exist a sequence of rational numbers that converges to it and there exist a sequence of irrational numbers that converges to it. What is the limit of this function as you approach x along either sequence?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Oct 2012
    From
    Israel
    Posts
    26

    Re: Non continuous function

    Hey I just wrote which Dirichlet function I meant.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,427
    Thanks
    1857

    Re: Non continuous function

    Okay, so what I said still applies. Take the limits as you approach x along a sequence of rational numbers and along a sequence of irrational numbers.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Mar 2010
    Posts
    993
    Thanks
    244

    Re: Non continuous function

    Or look at the interval (x-\delta,x+\delta) (using any value of \delta). Can you get |f(x)-f(x_0)|<\epsilon?

    - Hollywood
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. prove that every continuous function is separately continuous
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: November 23rd 2011, 04:57 AM
  2. Replies: 3
    Last Post: April 18th 2011, 09:19 AM
  3. Replies: 3
    Last Post: April 18th 2011, 08:24 AM
  4. Replies: 2
    Last Post: September 2nd 2010, 11:28 AM
  5. Continuous Function
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: April 16th 2009, 12:50 PM

Search Tags


/mathhelpforum @mathhelpforum