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:

Here x runs through the domain of f. In formulas, this can be expressed as follows:

For a function f defined on ${\displaystyle \mathbb {R} }$, the definition can be expressed in terms of the halo as follows: f is microcontinuous at ${\displaystyle c\in \mathbb {R} }$ if and only if ${\displaystyle f(hal(c))\subset hal(f(c))}$, 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 ${\displaystyle {\text{st}}\circ f}$ is constant of the halo of c, where "st" is the standard part function.

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