Editing Infinitesimal (section)

== Functions tending to zero ==
In a related but somewhat different sense, which evolved from the original definition of "infinitesimal" as an infinitely small quantity, the term has also been used to refer to a function tending to zero. More precisely, Loomis and Sternberg's ''Advanced Calculus'' defines the function class of infinitesimals, <math>\mathfrak{I}</math>, as a subset of functions <math>f:V\to W</math> between normed vector spaces by <blockquote><math>\mathfrak{I}(V,W) = \{f:V\to W\ |\ f(0)=0,
(\forall \epsilon>0) (\exists \delta>0)
\ \backepsilon\ ||\xi||<\delta\implies
||f(\xi)||<\epsilon\}</math>, </blockquote>as well as two related classes <math>\mathfrak{O},\mathfrak{o}</math> (see [[Big O notation|Big-O notation]]) by <blockquote><math>\mathfrak{O}(V,W) = \{f:V\to W\ |\ f(0)=0,\ (\exist r>0,c>0)\ \backepsilon\ 
||\xi||< r \implies ||f(\xi)||\leq c||
\xi||\}</math>, and</blockquote><blockquote><math>\mathfrak{o}(V,W) = \{f:V\to W\ |\ f(0)=0,\ \lim_{||\xi||\to 0} ||f(\xi)|| / ||\xi|| = 0\}</math>.<ref>{{Cite book|url=https://archive.org/details/LoomisL.H.SternbergS.AdvancedCalculusRevisedEditionJonesAndBartlett|title=Advanced Calculus|last1=Loomis|first1=Lynn Harold|last2=Sternberg|first2=Shlomo|publisher=World Scientific|year=2014|isbn=978-981-4583-92-3|location=Hackensack, N.J.|pages=138–142}}</ref></blockquote>The set inclusions <math>\mathfrak{o}(V,W)\subsetneq\mathfrak{O}(V,W)\subsetneq\mathfrak{I}(V,W)</math>generally hold. That the inclusions are proper is demonstrated by the real-valued functions of a real variable <math>f:x\mapsto |x|^{1/2}</math>, <math>g:x\mapsto x </math>, and <math>h:x\mapsto x^2 </math>:  <blockquote><math>f,g,h\in\mathfrak{I}(\mathbb{R},\mathbb{R}),\  g,h\in\mathfrak{O}(\mathbb{R},\mathbb{R}),\ 
h\in\mathfrak{o}(\mathbb{R},\mathbb{R})</math> but <math>f,g\notin\mathfrak{o}(\mathbb{R},\mathbb{R})</math> and <math>f\notin\mathfrak{O}(\mathbb{R},\mathbb{R})</math>.</blockquote>As an application of these definitions, a mapping <math>F:V\to W</math> between normed vector spaces is defined to be differentiable at <math>\alpha\in V</math> if there is a <math>T\in\mathrm{Hom}(V,W)</math> [i.e, a bounded linear map <math>V\to W</math>] such that <blockquote><math>[F(\alpha+\xi)-F(\alpha)]-T(\xi)\in \mathfrak{o}(V,W)</math></blockquote>in a neighborhood of <math>\alpha</math>. If such a map exists, it is unique; this map is called the ''differential'' and is denoted by <math>dF_\alpha</math>,<ref>This notation is not to be confused with the many other distinct usages of ''d'' in calculus that are all loosely related to the classical notion of the differential as "taking an infinitesimally small piece of something": ''(1)'' in the expression<math>\int f(x)\, d\alpha(x)</math>, <math>d\alpha(x)</math> indicates Riemann-Stieltjes integration with respect to the integrator function <math>\alpha</math>; ''(2)'' in the expression <math>\int f\, d\mu</math>, <math>d\mu</math> symbolizes Lebesgue integration with respect to a measure <math>\mu</math>; ''(3)'' in the expression <math>\int_{\mathbf{R}^n} f\; dV</math>, ''dV'' indicates integration with respect to volume; ''(4)'' in the expression <math>dx^{i_1}\wedge\cdots\wedge dx^{i_n}</math>, the letter ''d'' represents the exterior derivative operator, and so on....</ref> coinciding with the traditional notation for the classical (though logically flawed) notion of a differential as an infinitely small "piece" of ''F''. This definition represents a generalization of the usual definition of differentiability for vector-valued functions of (open subsets of) Euclidean spaces.