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If you are sure that $f$ is continuous and does not vanish in the integration domain, it is by no means necessary making use of regularization theory of distributions. Consider the initial integral: $$F:= \int_0^1 \mathrm{d}x\int_0^{1-x}\mathrm{d}y \frac{1}{f(x,y)+\mathrm{i}\epsilon}.$$ It can be re-written as: F:= \int_{T} ...