# Multiple Integrals for Functions Unbounded at Isolated Points

1. Nov 18, 2011

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. Nov 18, 2011

### mathman

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.

3. Nov 18, 2011

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??

4. Nov 19, 2011

### mathman

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

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