Microcontinuity

In <a href="/facts/Nonstandard_analysis/lEpNpWMB">nonstandard analysis</a>, a discipline within classical mathematics, microcontinuity (or S-continuity) of an <a href="/facts/Internal_function/IHp7mBaK">internal function</a> 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 <a href="/facts/Halo_(mathematics)/pJ3DspXx">halo</a> 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 <a href="/facts/Hyperreal_number/Kvf09qpo">hyperreals</a> 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 <a href="/facts/Standard_part_function/fTuL5IXj">standard part function</a>.

Microcontinuity open-in-new

Microcontinuity