(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that f(x)=[tex]\frac{1}{x^{2}}[/tex] is uniformly continuous on the set [1,[tex]\infty[/tex]) but not on the set (0,1].

2. Relevant equations

3. The attempt at a solution

I've been working at this for at least 2 hours now, possibly 3, and I can't say I really have much of any idea about it. I've primarily been looking at the fact that a function f: A[tex]\rightarrow[/tex] [tex]\Re[/tex] is not uniformly continuous iff [tex]\exists[/tex] some [tex]\epsilon_{0}[/tex] > 0 and sequences x_{n}and y_{n}where |x_{n}-y_{n}| [tex]\rightarrow[/tex] 0 but |f(x_{n})-f(y_{n})|[tex]\geq[/tex] [tex]\epsilon[/tex]_{0}.

My thought was to somehow say that, for part one, because all sequences must be [tex]\geq[/tex] 1, they must converge to the same thing and therefore |f(x_{n})-f(y_{n})|= 0 for some n, but I'm not sure if I can for various reasons. In fact, I can't even really come up with any sequences in (0,1] to show that the function is not uniformly continuous on that set. I can think of another way to do that one, but I really want to see some sequences to do it to prove that it works in that case.

Any help is greatly appreciated.

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# Homework Help: Uniform Continuity of 1/x^2 on various sets

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