Microcontinuity

In nonstandard 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:

if

x

≈

a

{\displaystyle x\approx a}

then

f

(

x

)

≈

f

(

a

)

{\displaystyle f(x)\approx f(a)}

.

For a function f defined on

R

{\displaystyle \mathbb {R} }

, the definition can be expressed in terms of the halo as follows: f is microcontinuous at

c

∈

R

{\displaystyle c\in \mathbb {R} }

if and only if

f

(

h

a

l

(

c

)

)

⊆

h

a

l

(

f

(

c

)

)

{\displaystyle f(hal(c))\subseteq 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

st

∘

f

{\displaystyle {\text{st}}\circ f}

is constant on 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.

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 nonstandard) of the natural extension I* of its domain I (see Davis, 1977, p.