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Using reinmann sum

  1. Mar 23, 2007 #1
    can we use rienmann sum to calculate the area that is enclosed between a part of a function and the Ox axes in the interval [a,b) if the function is not defined at b ?
     
  2. jcsd
  3. Mar 23, 2007 #2

    uart

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    Yes. An isolated point cannot change the value of the area under a function.

    For that matter neither can a large number or even a countably infinite number of isolated points change the value of an integral. (assuming of course that's there's no nasties like the Dirac Delta "function" involved).
     
  4. Mar 23, 2007 #3

    matt grime

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    Riemann sum. Not Reinmann. Adn to expand on Uart's post. The Riemann integral is the limit as epsilon tends to zero of the integrals [a,b-epsilon], when it exists.
     
  5. Mar 23, 2007 #4
    Yes you can, if the function is bounded on [a, b). And it is equal to the integral of the function [a,b].

    This is not true for the Reimann integral, which is why the Reimann integral is utterly worthless, in favor of Lebesgue.

    For example, the function:

    f(x) = {0 if x is irrational, 1 if x is rational}

    has only countably many discontinuities, but it not reimann integrable:frown:
     
  6. Mar 23, 2007 #5
    No, I'm pretty sure that function is discontinuous everywhere.
     
  7. Mar 23, 2007 #6

    matt grime

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    Even if the singularities were only at the rationals (and they aren't, as moo points out) they fail uart's restriciton to isolated singularities.
     
  8. Mar 23, 2007 #7

    HallsofIvy

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    Any bounded function, as long as the set of discontinuities has measure 0, is Riemann integrable. Any countable set has measure 0. As both Moo of Doom and matt grime said, the function you give is discontinuous everywhere. Its set of discontinuities has measure 1.
     
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