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
f(x) ≈ f(a)
For a function f defined on
R
c\inR
f(hal(c))\subseteqhal(f(c))
st\circf
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]
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. 96).
The real function
f(x)=\tfrac{1}{x}
a>0
\tfrac{1}{a}
\tfrac{1}{2a}
The function
f(x)=x2
R
H\inR*
e=\tfrac{1}{H}
Uniform convergence similarly admits a simplified definition in a hyperreal setting. Thus, a sequence
fn
*(x) | |
f | |
n |
f*(x)