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.
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.
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].
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?