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Math Help - continuity

  1. #1
    mms
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    continuity

    Let <br />
f:A \subset \mathbb{R} \to \mathbb{R}<br />
and r_n > 0
    a sequence such that <br />
r_n \to 0<br /> <br />
    Prove that f is continuos in \overline x
    if and only if

    <br />
s_n = \sup \left\{ {\left| {f\left( x \right) - f\left( {\overline x } \right)} \right|:\,\left| {x - \overline x } \right| \leqslant r_n } \right\}<br /> <br />
converges to zero

    thanks!
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Where are you stuck ?
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  3. #3
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    Use this fact, which is "easier" to prove (though more or less the same).

    f:X\to{Y} is continuous at x if and only if for every sequence \{x_n\}\in X such that x_n\to x, f(x_n)\to f(x).
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  4. #4
    Super Member Gamma's Avatar
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    Quote Originally Posted by putnam120 View Post
    Use this fact, which is "easier" to prove (though more or less the same).

    f:X\to{Y} is continuous at x if and only if for every sequence \{x_n\}\in X such that x_n\to x, f(x_n)\to f(x).
    Careful there, this is only an if and only if theorem if X is metrizable.

    In general it is only true that a function being continuous implies for every convergent sequence x_n \rightarrow x implies f(x_n)\rightarrow f(x).


    Obviously \mathbb{R} is metrizable, so it is okay to use here, but just want to make sure no passerby thinks this theorem is true in general as you have it stated
    Last edited by Gamma; August 5th 2009 at 06:22 PM. Reason: added last paragraph
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  5. #5
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    Oh thank. Guess I have just gotten used to always working with metric spaces that I no longer mention it.
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