Innonstandard analysis, a discipline within classical mathematics,microcontinuity (orS-continuity) of aninternal functionf at a pointa is defined as follows:

for allx infinitely close toa, the valuef(x) is infinitely close tof(a).

Herex runs through the domain off. In formulas, this can be expressed as follows:

if${\displaystyle x\approx a}$ then${\displaystyle f(x)\approx f(a)}$.

For a functionf defined on${\displaystyle \mathbb{R}}$, the definition can be expressed in terms of thehalo as follows:f is microcontinuous at${\displaystyle c\in \mathbb{R}}$ if and only if${\displaystyle f(hal(c))\subseteq hal(f(c))}$, where the natural extension off to thehyperreals is still denotedf. Alternatively, the property of microcontinuity atc can be expressed by stating that the composition${\displaystyle \text{st}\circ f}$ is constant on the halo ofc, where "st" is thestandard part function.

The modern property of continuity of a function was first defined by Bolzano in 1817. However, Bolzano's work was not noticed by the larger mathematical community until its rediscovery in Heine in the 1860s. Meanwhile,Cauchy's textbookCours d'Analyse defined continuity in 1821 usinginfinitesimals as above.^[1]

The property of microcontinuity is typically applied to the natural extensionf* of a real functionf. Thus,f defined on a real intervalI iscontinuous if and only iff* is microcontinuous at every point ofI. Meanwhile,f isuniformly continuous onI if and only iff* is microcontinuous at every point (standard and nonstandard) of the natural extensionI* of its domainI (see Davis, 1977, p. 96).

The real function${\displaystyle f(x)=\frac{1}{x}}$ on the open interval (0,1) is not uniformly continuous because the natural extensionf* off fails to be microcontinuous at aninfinitesimal${\displaystyle a>0}$. Indeed, for such ana, the valuesa and2a are infinitely close, but the values off*, namely${\displaystyle \frac{1}{a}}$ and${\displaystyle \frac{1}{2a}}$ are not infinitely close.

The function${\displaystyle f(x)=x^2}$ on${\displaystyle \mathbb{R}}$ is not uniformly continuous becausef* fails to be microcontinuous at an infinite point${\displaystyle H\in {\mathbb{R}}^{\ast }}$. Namely, setting${\displaystyle e=\frac{1}{H}}$ andK = H + e, one easily sees thatH andK are infinitely close butf*(H) andf*(K) are not infinitely close.

Uniform convergence similarly admits a simplified definition in a hyperreal setting. Thus, a sequence${\displaystyle f_n}$ converges tof uniformly if for allx in the domain off* and all infiniten,${\displaystyle f_n^{\ast }(x)}$ is infinitely close to${\displaystyle f^{\ast }(x)}$.

• Standard part function

• Martin Davis (1977) Applied nonstandard analysis. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney. xii+181 pp.ISBN 0-471-19897-8
• Gordon, E. I.; Kusraev, A. G.;Kutateladze, S. S.: Infinitesimal analysis. Updated and revised translation of the 2001 Russian original. Translated by Kutateladze. Mathematics and its Applications, 544. Kluwer Academic Publishers, Dordrecht, 2002.

• Nonstandard analysis
• Theory of continuous functions

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