International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 29

The guiding principle which leads to requiring that the functions ϕ above (traditionally called `test functions') should be well behaved is that correspondingly `wilder' behaviour can then be accommodated in the limiting behaviour of the while still keeping the integrals under control. Thus
To ensure further the continuity of functionals such as with respect to the test function ϕ as the go increasingly wild, very strong control will have to be exercised in the way in which a sequence of test functions will be said to converge towards a limiting ϕ: conditions will have to be imposed not only on the values of the functions , but also on those of all their derivatives. Hence, defining a strong enough topology on the space of test functions ϕ is an essential prerequisite to the development of a satisfactory theory of distributions.