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Homework Help: Uniform Continuity and Supremum

  1. Dec 14, 2011 #1
    Last edited: Dec 14, 2011
  2. jcsd
  3. Dec 14, 2011 #2


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    What kinds of functions have you tried and why do you think they aren't working?
  4. Dec 14, 2011 #3


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    Hint: That's a difference quotient.
  5. Dec 14, 2011 #4
    Further hint: find a function whose derivative is infinite/doesn't exist...
  6. Dec 14, 2011 #5
    Does x*sin(1/x) work since its derivative is undefined at x=0 which is in [-1,1]?
  7. Dec 14, 2011 #6

    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
  8. Dec 14, 2011 #7
    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.
  9. Dec 14, 2011 #8

    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)
  10. Dec 14, 2011 #9
    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].
  11. Dec 14, 2011 #10
    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.
  12. Dec 14, 2011 #11
    So it would just be sqrt(x+1) for the [-1,1] interval as another solution?
  13. Dec 14, 2011 #12
    That should do it.
  14. Dec 14, 2011 #13
    Last edited: Dec 14, 2011
  15. Dec 14, 2011 #14


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    It's not even true. There are functions that differentiable at x=0, that aren't even differentiable anywhere else.
  16. Dec 14, 2011 #15
    Sorry, there's a condition also that f: R-->R and must be differentiable on R
  17. Dec 14, 2011 #16


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    Then maybe use the mean value theorem?
  18. Dec 15, 2011 #17

    I like Serena

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    Just out of curiosity, renjean and chairbear, what is the reason you deleted your questions?
  19. Dec 15, 2011 #18
    It's because they are cheating. Please report these kind of things.
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