Microcontinuity

In non-standard analysis, a discipline within classical mathematics, microcontinuity (or S-continuity) of an internal function f at a point a is defined as follows:

for all x infinitely close to a, the value f(x) is infinitely close to f(a).

Here x runs through the domain of f. In formulas, this can be expressed as follows:

x\approx a

then

f(x)\approx f(a)

For a function f defined on $\mathbb {R}$ , the definition can be expressed in terms of the halo as follows: f is microcontinuous at $c\in {\mathbb {R}}$ if and only if $f(hal(c))\subset hal(f(c))$ , where the natural extension of f to the hyperreals is still denoted f. Alternatively, the property of microcontinuity at c can be expressed by stating that the composition ${\text{st}}\circ f$ is constant of the halo of c, where "st" is the standard part function.

History

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 textbook Cours d'Analyse defined continuity in 1821 using infinitesimals as above.^[1]

Continuity and uniform continuity

The property of microcontinuity is typically applied to the natural extension f* of a real function f. Thus, f defined on a real interval I is continuous if and only if f* is microcontinuous at every point of I. Meanwhile, f is uniformly continuous on I if and only if f* is microcontinuous at every point (standard and non-standard) of the natural extension I* of its domain I (see Davis, 1977, p. 96).

Example 1

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

Example 2

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

Uniform convergence

Uniform convergence similarly admits a simplified definition in a hyperreal setting. Thus, a sequence $f_{n}$ converges to f uniformly if for all x in the domain of f* and all infinite n, $f_{n}^{*}(x)$ is infinitely close to $f^{*}(x)$ .

Bibliography

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.

References

↑ Borovik, Alexandre; Katz, Mikhail G. (2011), "Who gave you the Cauchy--Weierstrass tale? The dual history of rigorous calculus", Foundations of Science, arXiv:1108.2885, doi:10.1007/s10699-011-9235-x .

Infinitesimals

History	Adequality Leibniz's notation Integral symbol Criticism of non-standard analysis The Analyst The Method of Mechanical Theorems Cavalieri's principle Method of indivisibles

Related branches	Non-standard analysis Non-standard calculus Internal set theory Synthetic differential geometry Constructive non-standard analysis Infinitesimal strain theory (physics)

Formalizations	Differentials Hyperreal numbers Dual numbers Surreal numbers

Individual concepts	Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of Continuity Overspill Microcontinuity Transcendental law of homogeneity

Scientists	Gottfried Wilhelm Leibniz Abraham Robinson Pierre de Fermat Augustin-Louis Cauchy Leonhard Euler

Textbooks	Analyse des Infiniment Petits Elementary Calculus Cours d'Analyse

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