# Homework Help: Uniform continuity with bounded functions

1. Jul 20, 2007

### 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.

2. Jul 20, 2007

### chaoseverlasting

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

3. Jul 20, 2007

### daniel_i_l

Yes, but it's not bounded in R.

4. Jul 20, 2007

### Dick

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

5. Jul 22, 2007

### 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.

6. Jul 23, 2007

### 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.

7. Jul 23, 2007