Results 1 to 3 of 3

Thread: Counterexample of two functions not continuous but composition is

  1. #1
    Junior Member
    Joined
    Jun 2011
    From
    Colorado, United States
    Posts
    56

    Counterexample of two functions not continuous but composition is

    Question:
    If f: D->E and g: E->F are not continuous on D and E, respectively, then the composition g $\displaystyle \circ$f is NOT continuous on D.

    I'm guessing False on this statement but can't come up with the counterexample needed.

    Any suggestions?

    Thank you.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    678
    Thanks
    32

    Re: Counterexample of two functions not continuous but composition is

    For example take $\displaystyle D=E=F =\mathbb R$ and $\displaystyle f(x)=g(x) = \begin{cases}1&\mbox{if }x\in\mathbb{Q}\\ 0&\mbox{if }x\in\mathbb{R}\setminus \mathbb{Q}.\end{cases}$
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1

    Re: Counterexample of two functions not continuous but composition is

    Quote Originally Posted by girdav View Post
    For example take $\displaystyle D=E=F =\mathbb R$ and $\displaystyle f(x)=g(x) = \begin{cases}1&\mbox{if }x\in\mathbb{Q}\\ 0&\mbox{if }x\in\mathbb{R}\setminus \mathbb{Q}.\end{cases}$
    For CountingPenguins:

    g(x) in girdav post is Dirichlet Function -- from Wolfram MathWorld
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. prove that the composition gof:A-> C is uniformly continuous
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Apr 9th 2011, 12:49 PM
  2. Composition Continuous
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: Feb 22nd 2010, 09:38 PM
  3. need a counterexample for Riemann Integrable functions problem
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Dec 5th 2009, 07:29 PM
  4. Replies: 0
    Last Post: Oct 14th 2009, 10:16 AM
  5. composition of functions
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: Feb 24th 2009, 11:46 AM

Search Tags


/mathhelpforum @mathhelpforum