Microcontinuity

In non-standard analysis, a discipline within classical mathematics, microcontinuity of a 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.

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 it 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).

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 defined continuity in 1821 using infinitesimals as above.^[1]

Example 1

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

Example 2

The function on is not uniformly continuous because f* fails to be microcontinuous at an infinite point . Namely, setting 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 converges to f uniformly if for all x in the domain of f* and all infinite n, is infinitely close to .

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

1. ^ 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 .