Consider the function

Now, , which is not bounded, so it is not uniformly continuous.

So pick , let , for every , we need , but how should I go about that? Thanks.

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- October 12th 2008, 12:08 PMtttcomraderNot Uniform Continuous?
Consider the function

Now, , which is not bounded, so it is not uniformly continuous.

So pick , let , for every , we need , but how should I go about that? Thanks. - October 12th 2008, 12:29 PMLaurent
" is not uniformly continuous" means:

such that and .

I let you get convinced yourself that a proof would consist in finding a sequence a couples such that and for some (or even , if possible).

In order to choose those couples, look at (or just imagine) what the graph looks like: the must be close but the must be way apart from each other.

is a possible start. - October 12th 2008, 04:50 PMtttcomrader
Thanks for the hint. So for the choice that , we would have , but any point right afterward would get larger as n increases.

Pick , then , but

Is this right? - October 12th 2008, 11:59 PMLaurent
No, because

*for every*you must have for some . This is why I said you must have .

You can write where is to be determined.

You have as . So that you must choose such that . For instance, (hence must be small enough), or (and any is adequate).

You can do a more explicit proof using the convexity inequality which holds as soon as . This yields to the same possibilities for the choice of . - October 16th 2008, 09:32 AMtttcomrader
Thanks for the help. But one thing I'm not clear is, why is it enough to show that implies ?

- October 16th 2008, 11:06 AMLaurent
Explicitating my first post as much as I can:

You know that " is not uniformly continuous" means:

such that and .

You obtain this by taking the logical contrary of the definition of "uniformly continuous".

Suppose we found a sequence of couples and an such that and, for all , .

Then, for every , there is an such that (because of the convergence to 0). And for this same , we have . As a consequence, we have realized the above definition of "not uniformly continuous" with in place of .