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Multiple Integrals for Functions Unbounded at Isolated Points

  1. Nov 18, 2011 #1
    In a recent homework assignment, I was asked to prodive a definition for ∫f(x) in the Region D, provided there was a discontinuity somewhere in the region. To define the integral, we merely removed a sphere centered on the discontinuity of radius δ>0 and found the limit of the integral as δ→0.

    My question is how would you provide a more generalized definition for a function that had multiple undefined points? if i had points (x1,y1) and (x2,y2) where the function was undefined.

    Would I somehow split the integral up into three parts?
  2. jcsd
  3. Nov 18, 2011 #2


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    Whatever trick you used can be used separately at each point in question. You would need a more sophisticated approach if there are an infinite number of points involved.
  4. Nov 18, 2011 #3
    but if the integral originally broke up the bounds to be one integral with (a, δ) and the second integal with(δ,b), with δ going to zero after the integral was evaluated, then how would the bounds look with more points??

    (a,δ1) , (δ1,δ2), (δ2,δ3), ....etc (δn, b) ??

    with each of those δ, going to zero??
  5. Nov 19, 2011 #4


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    Your notation is very confusing. The general idea is to treat each discontinuity separately.

    You use a, b, various δ's, without defining anything.
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