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.
" 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.
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 .
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 .