# Common Concepts - A Non-standard View

The motivation behind the development of the Non-standard analysis was to supply rigorous justification the use of the infinitesimals in calculus. Below I list a few basic definitions that demonstrate how that goal has been achieved. Subsequent pages will corroborate those definition. To avoid unnecessary formalities we shall assume that the functions referred to below are given by algebraic formulas so that they are automatically defined on the set of hyperreals whenever they are defined for the real numbers. More generally, it must be remembered that the sets of the reals (R) and the hyperreals (R*) are both models of the same mathematical theory (R). For that reason, the statements of one within that theory admit an interpretation as the statements of the other. Since the theory R* is an expansion of R with the axioms that define infinite and, hence, infinitesimal numbers, not all statements of R* admit an interpretation in R. This is the subject of the Transfer Principle.

### Limit of a sequence

A sequence {sn} has a limit L, lim sn = L iff sn ≈ L for all infinite n ∈ N* - N.

### Continuity of a function

Real function f is continuous at a real point a iff, for all x ≈ a, f(x) ≈ f(a).

### Uniform continuity

Real function f is uniformly continuous over a set A iff for all x ≈ y, x, y ∈ A, f(x) ≈ f(y).

### Open sets

Set A is open iff, for and real x, x∈A implies Mon(x)⊂A.

### Closed sets

Set A is closed iff, for any finite x, A∩Mon(x) ≠ Ø implies st(x) ∈ A.

### Compactness

Set A is compact iff x ∈ A implies st(x) exists and st(x) ∈ A.

1. J. M. Henle, E. M. Kleinberg, Infinitesimal Calculus, Dover, 2003
2. A. Robinson, Non-standard Analysis, Princeton University Press (Rev Sub edition), 1996

Back to What Is Infinity?