linguistic:voice_change, linguistic:translation, structural:summarization, information:classification
Infinitesimal delta functions

Cauchy used an infinitesimal α {\displaystyle \alpha } to write down a unit impulse, infinitely tall and narrow Dirac-type delta function δ α {\displaystyle \delta _{\alpha }} satisfying ∫ F (x) δ α (x) = F (0) {\displaystyle \int F(x)\delta _{\alpha }(x)=F(0)} in a number of articles in 1827, see Laugwitz (1989). Cauchy defined an infinitesimal in 1821 (Cours d'Analyse) in terms of a sequence tending to zero. Namely, such a null sequence becomes an infinitesimal in Cauchy's and Lazare Carnot 's terminology.

Modern set-theoretic approaches allow one to define infinitesimals via the ultrapower construction, where a null sequence becomes an infinitesimal in the sense of an equivalence class modulo a relation defined in terms of a suitable ultrafilter. The article by Yamashita (2007) contains bibliography on modern Dirac delta functions in the context of an infinitesimal-enriched continuum provided by the hyperreals.