Let be continuous and bounded on function such that
Does it follows that is uniformly continuous on ?
Solution. Suppose for example that is bounded and that is not uniformly continuous. Then one can find such that there exists a sequence satisfying and . Now, take such that
So if satisfies and , then it is easy to see that:
for . Hence
which shows that cannot be bounded, a contradiction.
More. Suppose continous function on
Suppose there exist such that
each time it is possible (in french “chaque fois que cela a un sens”)
1/ prove is bounded on
2/ Suppose with
Prove for any
Prove there exist such that for any small