What is "microcontinuity"

In non-standard 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 then f(x) ≈ f(a).

For a function f defined on R, the definition can be expressed in terms of the halo as follows: f is microcontinuous at c ∈ R if and only if f(hal(c)) ⊂ 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 is constant of the halo of c, where "st" is the standard part function.