Recognitions:
Gold Member

## uniform continuity with bounded functions

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.
 Dunno if this helps, but for the second one, y=x^2 is an example whose slope increases as x approaches infinity.
 Recognitions: Gold Member Yes, but it's not bounded in R.

Recognitions:
Homework Help

## uniform continuity with bounded functions

Doesn't your sin(x^2) example show both statements are false?
 Recognitions: Gold Member 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.

Recognitions:
Homework Help