- Euler's vision of a generalized concept of function was a forerunner of the modern concept of distribution, and his efforts to give meaning to divergent series eventually led to the concepts of asymptotic series, summability, and distributional convergence. The introduction of such suitable abstract concepts does not automatically prevent mistakes or inconsistencies resulting from careless formal reasoning. We deal with a cluster of such issues associated with the occurrence of a distributional singularity on the boundary of a domain of integration. Apparent paradoxes are resolved by introducing new classes of test functions and distributions adapted to the problems at hand; one can regard the construction as attributing internal structure to boundary points. 2007 by IJAMAS, CESER.