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Homework Help: Uniform convergence and continuity

  1. Jun 8, 2010 #1
    1.kn (x) = 0 for x ≤ n
    x − n, x ≥ n,
    Is kn(x) uniformly convergent on R?

    I can show that it is uniformly convergent on any closed bounded interval [a,b], but I don't think it is on R. Could anyone please give me some hints how to prove it?

    2.Fix 0 < η < 1. Suppose now that h : [0, 1] → R is continuous. Prove that the series
    t(x) = ∑ x^n h(x^n ) is uniformly convergent on [0, η]. Deduce that t(x) is continuous.

    I'm not sure how to treat h(x^n) here, since it's not bounded. Could anyone help me figure it out?

    Any help is appreciated!
     
  2. jcsd
  3. Jun 8, 2010 #2

    lanedance

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    for the first, how about a slightly simpler case first... can you show whether y=x is uniformly convergent?
     
  4. Jun 8, 2010 #3

    lanedance

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    for the 2nd, i would start by considering the defintion of uniform convergence

    so start with any e>0 and look at |t(x+e) - t(x)|

    i haven't tried it yet, but i think the idea is the multiplication of x^n and the continuity of h(x) should do it...
     
  5. Jun 8, 2010 #4
    Thanks for your reply. I think clearly y=x is not uniformly convergent, so I guess kn(x) isn't either?
    About the second one, I tried to work with the epsilon-delta definition, but the result seemed still depend on n. Could you please be a bit more specific how you would do it?
     
  6. Jun 8, 2010 #5
    Is the first question asking whether kn(x) is uniformly continuous? If not, is it asking whether the sequence kn(x) is uniformly convergent to the zero function? For the second question, are you familiar with the theorem that says a continuous function on a compact set achieves its maximum and minimum?
     
  7. Jun 8, 2010 #6
    Thanks, I got it!
     
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