uniform continuity with bounded functions

daniel_i_l

1. The problem statement, all variables and given/known data
True or false:
1)If f is bounded in R and is uniformly continues in every finate segment of R then it's uniformly continues for all R.
2)If f is continues and bounded in R then it's uniformly continues in R.


2. Relevant equations



3. The attempt at a solution

1) If we know that up to any x the function us UC in [0,x] and which means that the set A = {x | [0,x] in UC} has no upper bound, then does that mean that f is UC for {0,infinity) ? Why do I need the fact that they're bounded?
2) I think that the answer is no: can't we find some function whose slope increases as x goes to infinity? for example sin(x^2)?

In both of the questions I felt that I didn't have an intuitive way to combine the UC and the boundedness. Can anyone give me some directions?
Thanks.

chaoseverlasting

Dunno if this helps, but for the second one, y=x^2 is an example whose slope increases as x approaches infinity.

daniel_i_l

Yes, but it's not bounded in R.

Dick

Doesn't your sin(x^2) example show both statements are false?

daniel_i_l

Hmm, It looks like it does, do I guess that what I said in (1) is wrong.
But can you give me some general advice on how to approach problems dealing with bounded UC functions?
Thanks.

Dick

Nothing much more than you probably already know. If a function is differentiable UC means bounded derivative. So if I want a counterexample to 1) I look first at differentiable functions to see if I can find one that is bounded, but has unbounded derivative. sin(x^2) does nicely.

daniel_i_l

Thanks for your help.