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Is f integrable in [-2,2]?

  1. Aug 5, 2007 #1


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    1. The problem statement, all variables and given/known data
    f is defined by: for x=<0 f(x) = sin(x). for x>0 f(x) = sin(1/x).
    Is f integrable in [-2,2]?

    2. Relevant equations

    3. The attempt at a solution

    I think that the answer is yes because f is continues for all x in [-2,2] except for a finite amount of points (x=0). Is that right? It just seems weird that a function as chaotic as sin(1/x) could be integrable around 0.
  2. jcsd
  3. Aug 5, 2007 #2


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    It looks like you are right.

    As you already remarked, the problematic point is x = 0.
    Actually, the question reduces to: is sin(1/x) integrable on [0, a] (with a > 0).
    So is it integrable on [[itex]\epsilon[/itex], 2] and what happens if you take [itex]\epsilon \downarrow 0[/itex]?
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