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Math Help - Composite functions continuity theorem

  1. #1
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    Composite functions continuity theorem

    g: R->R

    g(x) = 2 if x not equals 1
    g(x) = 0 if x equals 1

    f(x) = x +1 for R

    "Verifiy" (sorry my english) that lim x-> 0 (gof)(x) is not equal lim x->0 (gof)(0)

    If there is a contradiction with the continuity theorem of composite function, give a justification why.



    ------------------------------------------------------------------

    Firs part:

    lim x->0 (gof)(x) = lim x->0 g(x+1) = g(1) = 2
    lim x->0(gof)(0) = g(0+1) lim x->0 g(1) = 0

    ??

    and now what is continuity theorem of composite function?
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  2. #2
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    Re: Composite functions continuity theorem

    Quote Originally Posted by Fabio010 View Post
    g: R->R
    g(x) = 2 if x not equals 1
    g(x) = 0 if x equals 1
    f(x) = x +1 for R
    "Verifiy" (sorry my english) that lim x-> 0 (gof)(x) is not equal lim x->0 (gof)(0)
    If there is a contradiction with the continuity theorem of composite function, give a justification why.
    lim x->0 (gof)(x) = lim x->0 g(x+1) = g(1) = 2
    lim x->0(gof)(0) = g(0+1) lim x->0 g(1) = 0
    The parts in red are not the way it is usually done.

    \lim _{x \to 0} g \circ f(x) = 2 BUT g \circ f(0) = 0\ne 2 shows that (gof) is not continuous at x=0~.
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  3. #3
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    Re: Composite functions continuity theorem

    Hum so gof is not continuous in x=0.

    But that contradicts the theorem of continuity?

    The only thing i know is that:

    if ( f is continuous in a and g is continuous in a then gof if continuouse in a)
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  4. #4
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    Re: Composite functions continuity theorem

    Quote Originally Posted by Fabio010 View Post
    Hum so gof is not continuous in x=0.
    But that contradicts the theorem of continuity?
    But in this case g is not continuous at x=1.
    \lim _{x \to 1} g(x) = 2 but g(1)=0.

    Quote Originally Posted by Fabio010 View Post
    if ( f is continuous in a and g is continuous in a then gof if continuouse in a)
    That should be g is continuous at f(a).
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