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Math Help - log x uniformly continuous

  1. #1
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    log x uniformly continuous

    having trouble proving log x being/not being uniformly continuous on:
    a. (x ϵ [1,∞))

    b. (x ϵ (0,1))

    Thank you again! and what is the best way about going about choosing a delta for these types of problems, I'm completely lost when it comes to proofs.
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  2. #2
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    Quote Originally Posted by ml692787 View Post
    having trouble proving log x being/not being uniformly continuous on:
    a. (x ϵ [1,∞))

    b. (x ϵ (0,1))

    Thank you again! and what is the best way about going about choosing a delta for these types of problems, I'm completely lost when it comes to proofs.
    (a) The derivative of the function \log x is bounded on (1.\infty). So what?

    (b) Assume that \log x was uniformly continous on (0,1) then any Cauchy sequence x_n in this interval implies f(x_n) is too a Cauchy sequence. Consider x_n = \frac{1}{n+1}. This is a sequence in (0,1) which is Cauchy. But yet f\left( \frac{1}{n+1} \right)  = - \log(n+1) is not because this gets larger and larger without bound.
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