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What functions are integrable?

  1. May 8, 2007 #1
    what functions are integrable?
     
  2. jcsd
  3. May 8, 2007 #2
    if f(x) is continuous on [a,b], then f(x) is integrable of [a,b]
     
  4. May 8, 2007 #3
    please explain completely and other situations
     
  5. May 8, 2007 #4
    What exactly do you mean? What kind of answer are you looking for?
     
  6. May 8, 2007 #5
    Essentially a function is integrable if it is bounded, and discontinuous on at most a set of measure zero.

    I suspect you are really asking: when can the anti-derivative of a function be expressed in terms of sums, products, powers, exponentials, trig functions, and the inverses of such. The conditions are not hard to state, but they are not interesting either.
     
  7. May 9, 2007 #6
    I think a function is integrable if it be dic-continus on countable point
     
  8. May 9, 2007 #7
    integral

    when a function is not continuos how can it be integrable?
     
  9. May 9, 2007 #8
    Not really treating this rigorously, but the idea is simple. Say f(x) = x at all x except when x=1. Say f(1) = 5.

    Then f is discontinuous at the point x=1. But if you interpret the integral as area under the graph, then it is intuitively clear that the integral of f from say, x=0 to x=2, is the same as that of the integral of the identity function from x=0 to x=2. The point at x=1 does not contribute to the area.

    So as Crosson said, essentially, a function is integrable if it is bounded, and discontinuous on at most a set of measure zero.
     
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