# Homework Help: Uniform Continuity and Supremum

1. Dec 14, 2011

### renjean

thanks!!!!

Last edited: Dec 14, 2011
2. Dec 14, 2011

### Dick

What kinds of functions have you tried and why do you think they aren't working?

3. Dec 14, 2011

### LCKurtz

Hint: That's a difference quotient.

4. Dec 14, 2011

### micromass

Further hint: find a function whose derivative is infinite/doesn't exist...

5. Dec 14, 2011

### renjean

Does x*sin(1/x) work since its derivative is undefined at x=0 which is in [-1,1]?

6. Dec 14, 2011

### micromass

Well,

1) is it uniform continuous??
2) Do the difference quotients go to infinity??

Just saying that the derivative is undefined isn't really enough...

PS There is a much easier example

7. Dec 14, 2011

### chairbear

I thought that any continuous function on a closed and bounded interval is also uniformly continuous. And in the case of x*sin(1/x) the derivative appears to go to infinity at x=0.

8. Dec 14, 2011

### micromass

Correct.

Not really. Rather, the derivative does not exist (since it oscillates too much). Nevertheless the supremum you mention does indeed go to infinity. (you might want to give a further proof if it is not clear)

9. Dec 14, 2011

### chairbear

I feel silly for making things more complicated than necessary. What was the easier example you had in mind? I was thinking square root x would work if the interval was [0,1].

10. Dec 14, 2011

### micromass

Don't feel silly. Your example is very elegant.

The square root is indeed the one I had in mind. You just need to modify it a bit.

11. Dec 14, 2011

### chairbear

So it would just be sqrt(x+1) for the [-1,1] interval as another solution?

12. Dec 14, 2011

### micromass

That should do it.

13. Dec 14, 2011

### chairbear

thanks

Last edited: Dec 14, 2011
14. Dec 14, 2011

### Dick

It's not even true. There are functions that differentiable at x=0, that aren't even differentiable anywhere else.

15. Dec 14, 2011

### chairbear

Sorry, there's a condition also that f: R-->R and must be differentiable on R

16. Dec 14, 2011

### Dick

Then maybe use the mean value theorem?

17. Dec 15, 2011

### I like Serena

Just out of curiosity, renjean and chairbear, what is the reason you deleted your questions?

18. Dec 15, 2011

### micromass

It's because they are cheating. Please report these kind of things.