Consider the following statements:
1.) If is uniformly continuous, then is bounded.
2.) If is uniformly continuous, and also if is bounded, then is bounded.
a) Is obviously false. Consider
Hints for b)
Uniformly continuous functions map Cauchy sequences into Cauchy sequences.
If E is closed the result in immediate.
You may try proof by contradiction.
Hints for b)
Uniformly continuous functions map Cauchy sequences into Cauchy sequences.
If E is closed the result in immediate.
You may try proof by contradiction.
That hint for b helped a lot! Thanks a lot, I figured that out. But, for a) (AKA 1), I still don't see how that disproves the statement. Mind elaborating?
That hint for b helped a lot! Thanks a lot, I figured that out. But, for a) (AKA 1), I still don't see how that disproves the statement. Mind elaborating?
the real-valued function f(x) = x is uniformly continuous, but if we take the domain of that function to be the reals, the range is unbounded