evenly continues question

• Aug 12th 2011, 10:06 AM
transgalactic
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
?
• Aug 12th 2011, 11:06 AM
SammyS
Re: uniformly continuous question
Quote:

Originally Posted by transgalactic
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 ?
• Aug 12th 2011, 11:08 AM
transgalactic
Re: evenly continues question
yes
• Aug 12th 2011, 03:39 PM
Plato
Re: evenly continues question
Quote:

Originally Posted by transgalactic
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.