# Thread: log x uniformly continuous

1. ## 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.

2. Originally Posted by ml692787
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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# log x uniformly continuous

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