Editing Nonstandard calculus (section)

==Uniform continuity==
A function ''f'' on an interval ''I'' is [[uniform continuity|uniformly continuous]] if its natural extension ''f''* in ''I''* has the following property:<ref>Keisler, Foundations of Infinitesimal Calculus ('07), p.&nbsp;45</ref>

for every pair of hyperreals ''x'' and ''y'' in ''I''*, if <math>x\approx y</math> then <math>f^*(x)\approx f^*(y)</math>.

In terms of microcontinuity defined in the previous section, this can be stated as follows: a real function is uniformly continuous if its natural extension f* is microcontinuous at every point of the domain of f*.

This definition has a reduced quantifier complexity when compared with the standard [[(ε, δ)-definition of limit|(ε,&nbsp;δ)-definition]].  Namely, the epsilon-delta definition of uniform continuity requires four quantifiers, while the infinitesimal definition requires only two quantifiers.  It has the same quantifier complexity as the definition of uniform continuity in terms of ''sequences'' in standard calculus, which however is not expressible in the [[first-order logic|first-order language]] of the real numbers.

The hyperreal definition can be illustrated by the following three examples.

Example 1: a function ''f'' is uniformly continuous on the semi-open interval (0,1], if and only if its natural extension f* is microcontinuous (in the sense of the formula above) at every positive infinitesimal, in addition to continuity at the standard points of the interval.

Example 2: a function ''f'' is uniformly continuous on the semi-open interval [0,∞) if and only if it is continuous at the standard points of the interval, and in addition, the natural extension ''f''* is microcontinuous at every positive infinite hyperreal point.

Example 3: similarly, the failure of uniform continuity for the squaring function

:<math>x^2</math>

is due to the absence of microcontinuity at a single infinite hyperreal point.

Concerning quantifier complexity, the following remarks were made by [[Kevin Houston (mathematician)|Kevin Houston]]:<ref>[[Kevin Houston (mathematician)|Kevin Houston]], How to Think Like a Mathematician, {{ISBN|978-0-521-71978-0}}</ref>

:The number of quantifiers in a mathematical statement gives a rough measure of the statement’s complexity. Statements involving three or more quantifiers can be difficult to understand. This is the main reason why it is hard to understand the rigorous definitions of limit, convergence, continuity and differentiability in analysis as they have many quantifiers.  In fact, it is the alternation of the <math>\forall</math> and <math>\exists</math> that causes the complexity.

[[Andreas Blass]] wrote as follows:
:Often ... the nonstandard definition of a concept is simpler than the standard definition (both intuitively simpler and simpler in a technical sense, such as quantifiers over lower types or fewer alternations of quantifiers).<ref>{{citation|first1=Andreas|last1=Blass
 |author1-link = Andreas Blass
 |title=Review: Martin Davis, Applied nonstandard analysis, and K. D. Stroyan and W. A. J. Luxemburg, Introduction to the theory of infinitesimals, and H. Jerome Keisler, Foundations of infinitesimal calculus|journal=Bull. Amer. Math. Soc.
 |volume=84
 |number=1
 |year=1978
 |pages=34–41|url=https://www.ams.org/journals/bull/1978-84-01/S0002-9904-1978-14401-2/home.html |doi=10.1090/S0002-9904-1978-14401-2|doi-access=free
 }}, p. 37.</ref>