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Math Help - Continuous Function Question

  1. #1
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    Continuous Function Question

    Suppose a function fa,b) -> R is continuous and f(r) = 0 for each rational number r in (a,b). Show that f(x) = 0 for each x element (a,b).

    So by the definition of continuous every sequence (xn) element dom(f) converges to x0 and limf(xn) = f(x0). Then, xn = 0 must converge to x0. Then, limf(0) = lim0 = 0 = f(x0).

    I was wondering if this was correct logic and if there other xn sequences that could be used.

    Thanks
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  2. #2
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    Quote Originally Posted by tbyou87 View Post
    Suppose a function fa,b) -> R is continuous and f(r) = 0 for each rational number r in (a,b). Show that f(x) = 0 for each x element (a,b).
    So by the definition of continuous every sequence (xn) element dom(f) converges to x0 and limf(xn) = f(x0). Then, xn = 0 must converge to x0. Then, limf(0) = lim0 = 0 = f(x0).
    I was wondering if this was correct logic and if there other xn sequences that could be used.
    Yes the logic is correct. Suppose that a < x_0  < b\quad \& \quad f\left( {x_0 } \right) \ne 0.
    A contradiction will follow.
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  3. #3
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    If you were doing it by contradiction, does this make sense:

    Assume f(x0) not equal 0 and a<x0<b.
    By definition of continuity, lim (xn) = x0 and lim f(xn) = f(x0).
    Since a<r<b, lim (xn) = r and lim f(xn) = f(r) = 0.
    Contradiction.

    Thanks
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  4. #4
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    Once again you have the right idea. But you need more detail.
    Suppose that f(x_0 ) \ne 0. Every real number is the limit of a sequence of rational numbers.
    So there is a sequence of rational numbers in (a,b) and \left( {r_n } \right) \to x_0 . But \left( {\forall n} \right)\left[ {f\left( {r_n } \right) = 0} \right].
    Is that possible if f(x_0 ) \ne 0 and \left( {f\left( {r_n } \right)} \right) \to f\left( {x_0 } \right) by continuity?
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  5. #5
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    No because continuity states that (f(rn)) -> f(x0) = f(r) = 0.
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  6. #6
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    Quote Originally Posted by tbyou87 View Post
    No because continuity states that (f(rn)) -> f(x0) = f(r) = 0.
    Now you understand the the whole idea.
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  7. #7
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    Can either of you help me with my limits problem? Its posted a few threads below this one. I need answers by tonite, so i really need help! Thanks.
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