1. 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

2. 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.

3. Thanks HallsofIvy