What kinds of functions have you tried and why do you think they aren't working?
Hint: That's a difference quotient.
Further hint: find a function whose derivative is infinite/doesn't exist...
Does x*sin(1/x) work since its derivative is undefined at x=0 which is in [-1,1]?
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
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.
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)
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].
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.
So it would just be sqrt(x+1) for the [-1,1] interval as another solution?
That should do it.
It's not even true. There are functions that differentiable at x=0, that aren't even differentiable anywhere else.
Sorry, there's a condition also that f: R-->R and must be differentiable on R
Then maybe use the mean value theorem?
Just out of curiosity, renjean and chairbear, what is the reason you deleted your questions?
It's because they are cheating. Please report these kind of things.
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