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Math Help - evenly continues question

  1. #1
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    evenly continues question

    f is continues in (a,b)
    lim_{x->b^-}f(x_0)=-\infty
    prove that f is not "evenly continues"
    ??
    i dont know the proper term.

    basicly we need to prove that there is \epsilon >0 so for every \delta >0 there is x for which |y_0-x_0|<\delta and |f(y_0)-f(x_0)|>=\epsilon

    if we choose \epsilon=1
    also we can use the limit
    ?
    Last edited by transgalactic; August 12th 2011 at 10:09 AM.
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  2. #2
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    Re: uniformly continuous question

    Quote Originally Posted by transgalactic View Post
    f is continues in (a,b)
    lim_{x->b^-}f(x_0)=-\infty
    prove that f is not "evenly continuous"
    ??
    i don't know the proper term.

    basicly we need to prove that there is \epsilon >0 so for every \delta >0 there is x for which |x-x_0|<\delta and |f-f(x_0)|>=\epsilon

    if we choose \epsilon=1
    also we can use the limit
    ?
    Do you mean: 'uniformly' continuous ?
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  3. #3
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    Re: evenly continues question

    yes
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  4. #4
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    Re: evenly continues question

    Quote Originally Posted by transgalactic View Post
    f is continues in (a,b) \lim_{x->b^-}f(x)=-\infty
    prove that f is not "'uniformly' continuous"???
    Do you know this theorem: If f is uniformly' continuous on a set S and (s_n) is a Cauchy Sequence in S then f(s_n) is a Cauchy Sequence ?
    You can use proof by contradiction. There is a positive integer K such that b-\frac{1}{K}\in (a,b).
    Define b_n=b-\frac{1}{K+n}. Now apply the theorem.
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