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Uniform continuity with bounded functions

  1. Jul 20, 2007 #1

    daniel_i_l

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    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. jcsd
  3. Jul 20, 2007 #2
    Dunno if this helps, but for the second one, y=x^2 is an example whose slope increases as x approaches infinity.
     
  4. Jul 20, 2007 #3

    daniel_i_l

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    Yes, but it's not bounded in R.
     
  5. Jul 20, 2007 #4

    Dick

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    Doesn't your sin(x^2) example show both statements are false?
     
  6. Jul 22, 2007 #5

    daniel_i_l

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    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.
     
  7. Jul 23, 2007 #6

    Dick

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    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.
     
  8. Jul 23, 2007 #7

    daniel_i_l

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    Thanks for your help.
     
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