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Math Help - Oscillation of a Function

  1. #1
    Super Member Bernhard's Avatar
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    Oscillation of a Function

    I am trying to get a good understanding of the concepts of lim sup, lim inf and oscillation of a function. I am lacking a source of good examples of these concepts. Can anyone advise me of a text or web site that provides some actual examples of finding lim sup, lim inf and oscillation (at a point) of a function.

    A particular problem.

    I wish to rigorously prove the following statement from Wikipedia regarding the oscillation of a function:

    sin (1/x) has oscillation 2 at x = 0, and 0 elsewhere.

    Can anyone help with a proof to give me an idea of the method for rigourously establishing the oscillation in the above case..

    Bernhard
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  2. #2
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    The "oscillation" of function f on interval [a, b] is sup(f)- inf(f) where where the sup and inf are for all values of x in the interval. The "oscillation" of f at a point, p, is the limit of the oscillation over intervals that include p as their length goes to 0. It should be obvious that the "oscillation" of a continuous function at a point is 0.

    Given any \epsilon> 0, for x between -\epsilon and \epsilon, 1/x takes on all values larger than 1/\epsilon and less than -1/\epsilon. That includes, of course, multiples of \pi so sin(1/x) takes on values of 1 and -1 in that interval and the "oscillation" in that interval is 1-(-1)= 2. The limit of that, as \epsilon goes to 0, is, of course, 2. That's why the oscillation of f(1/x) at x=0 is 2.

    Since f(1/x) is continuous at any non-zero x, its oscillation is 0 there.
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  3. #3
    Super Member Bernhard's Avatar
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    Smile

    Thanks HallsofIvy

    Your help is much appreciated!

    Bernhard
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