About: Microcontinuity

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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 then .

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