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Thread: 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 $\displaystyle \log x$ is bounded on $\displaystyle (1.\infty)$. So what?

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