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Math Help - Counterexample of two functions not continuous but composition is

  1. #1
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    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 \circf 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.
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  2. #2
    Super Member girdav's Avatar
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    Re: Counterexample of two functions not continuous but composition is

    For example take D=E=F =\mathbb R and 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}
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Counterexample of two functions not continuous but composition is

    Quote Originally Posted by girdav View Post
    For example take D=E=F =\mathbb R and 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
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