Let and let be defined by . Prove that is uniformly continuous on the interval , but is not uniformly continuous on .

I am lost on this problem. Any help is appreciated. Thanks in advance.

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- Mar 3rd 2009, 01:38 PMeskimo343uniformly continuous function
Let and let be defined by . Prove that is uniformly continuous on the interval , but is not uniformly continuous on .

I am lost on this problem. Any help is appreciated. Thanks in advance. - Mar 3rd 2009, 01:57 PMHallsofIvy
What do you have to work with? For example there is a theorem that says if a function is continuous on a compact set, then it is uniformly continuous there. If you do not have that theorem, Think about how you would prove continuity: If you want , then you want . How would you choose ? Show that if you can pick independent of but if can be arbitrarily close to 1, you can't.

- Mar 4th 2009, 08:25 PMeskimo343
- Mar 5th 2009, 12:31 AMOpalg
If a function has a bounded derivative on some interval then the function is uniformly continuous on that interval. (That is an easy consequence of the mean value theorem: if then for some c between x and y. So if then .)

The function has a bounded derivative on [a,2] if a>1. But as x→1, f'(x) becomes infinite. That means that you can find point x, y close to 1 such that |y-x| is very small but |f(y)–f(x)| is very large. So f is not uniformly continuous on (1,2]. - Mar 5th 2009, 09:58 AMeskimo343
We have not seen that result yet, so I am trying to prove this

__just__using the definition of uniform continuity. So, I did this:

Let . We assume that . Then So, I am really just wondering if this works to prove uniform continuity on . Does this show uniform continuity on just using the definition of uniform continuity? - Mar 6th 2009, 12:10 AMOpalg