5. Generalized numerical functions (distributions)

The notion of generalized functions extends that of ordinary functions. This notion appeared implicitly with Dirac's "function" (1926) and even earlier with Green's function (1828). The notion of a function defined almost everywhere can even be considered as explicitly the first modern precursor (Lebesgue, 1904).

Whereas an ordinary function f acts on the points of its definition domain by associating with a point x of this domain a value f(x) of its value domain, the alternative idea is to take into account the values of f at points in a neighborhood of the point x and to consider a generalized function T (in this article a distribution) as an operator on particular functions called test functions.

Distribution theory extends the notion of derivation to all locally integrable functions and beyond. It was introduced in 1936 by the Russian...