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A question about limit superior for function

  1. Aug 26, 2010 #1
    It is well known that limit of function can be converted to limit of sequence. I wonder if it still holds for limit superior of function. This problem is formulated as follows: For function [tex]f:\mathbb R\rightarrow\mathbb R[/tex] and [tex]a\in\mathbb R[/tex], define [tex]{\lim\sup}\limits_{x\to a}f(x)[/tex] to be [tex]\inf\limits_{\delta>0}(\sup\limits_{0<|x-a|<\delta}f(x))[/tex]. Can we have [tex]{\lim\sup}\limits_{x\to a}f(x)=c[/tex] iff [tex]{\lim\sup}\limits_{n\to\infty}f(x_n)=c[/tex] for any sequence [tex]<x_n>[/tex] satisfying 1)[tex]x_n\in\mathbb R[/tex], 2)[tex]x_n\to a[/tex] and 3)[tex]x_n\ne a[/tex]. I have no idea how to prove it, can you help me? Thanks!
     
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  3. Aug 26, 2010 #2

    mathman

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    It is not true as you stated it. The limsup (x->a) ≤ c for any sequence and = c for at least one sequence.
     
  4. Aug 31, 2010 #3
    I cannot understand your reply, could you please explain in more detail? Thanks.
     
  5. Aug 31, 2010 #4

    mathman

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    Example f(x)=1 for x rational, f(x)=0 for x irrational. Let a=0, limsup(x->a) f(x)=1. Take any sequence (xk) of irrational numbers converging to a, limsup f(xk)=0.
     
  6. Sep 1, 2010 #5
    A great example, I got it! Thank you!
    My original intention is to try to establish the statement "both lim sup f(x) and lim inf f(x) exist and equal c (possibly [tex]\pm\infty[/tex]) iff lim f(x) exists and equals c" from analogical statement for sequence. But now this approach is not feasible. I then proved the above statement for functions by definition. Thank you again, mathman!
     
  7. Sep 1, 2010 #6

    Office_Shredder

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    This is only true for functions that are continuous at the point. So it's not surprising that a limsup styled in the same manner would fail for a function that is everywhere discontinuous
     
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