Function from the limited hyperreal to the real numbers
In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every such hyperreal formula_1, the unique real formula_2 infinitely close to it, i.e. formula_3 is infinitesimal. As such, it is a mathematical implementation of the historical concept of adequality introduced by Pierre de Fermat, as well as Leibniz's Transcendental law of homogeneity.
The standard part function was first defined by Abraham Robinson who used the notation formula_4 for the standard part of a hyperreal formula_1 (see Robinson 1974). This concept plays a key role in defining the concepts of the calculus, such as continuity, the derivative, and the integral, in nonstandard analysis. The latter theory is a rigorous formalization of calculations with infinitesimals. The standard part of "x" is sometimes referred to as its shadow.
Definition.
Nonstandard analysis deals primarily with the pair formula_6, where the hyperreals formula_7 are an ordered field extension of the reals formula_8, and contain infinitesimals, in addition to the reals. In the hyperreal line every real number has a collection of numbers (called a monad, or halo) of hyperreals infinitely close to it. The standard part function associates to a hyperreal "x", the unique standard real number "x"0 that is infinitely close to it. The relationship is expressed symbolically by writing
formula_9
The standard part of any infinitesimal is 0. Thus if "N" is an infinite hypernatural, then 1/"N" is infinitesimal, and st(1/"N") = 0.
If a hyperreal formula_10 is represented by a Cauchy sequence formula_11 in the ultrapower construction, then 
formula_12
More generally, each finite formula_13 defines a Dedekind cut on the subset formula_14 (via the total order on formula_15) and the corresponding real number is the standard part of "u".
Not internal.
The standard part function "st" is not defined by an internal set. There are several ways of explaining this. Perhaps the simplest is that its domain L, which is the collection of limited (i.e. finite) hyperreals, is not an internal set. Namely, since L is bounded (by any infinite hypernatural, for instance), L would have to have a least upper bound if L were internal, but L doesn't have a least upper bound. Alternatively, the range of "st" is formula_16, which is not internal; in fact every internal set in formula_7 that is a subset of formula_8 is necessarily "finite".
Applications.
All the traditional notions of calculus can be expressed in terms of the standard part function, as follows.
Derivative.
The standard part function is used to define the derivative of a function "f". If "f" is a real function, and "h" is infinitesimal, and if "f"′("x") exists, then
formula_19
Alternatively, if formula_20, one takes an infinitesimal increment formula_21, and computes the corresponding formula_22. One forms the ratio formula_23. The derivative is then defined as the standard part of the ratio:
formula_24
Integral.
Given a function formula_25 on formula_26, one defines the integral formula_27 as the standard part of an infinite Riemann sum formula_28 when the value of formula_21 is taken to be infinitesimal, exploiting a hyperfinite partition of the interval ["a","b"].
Limit.
Given a sequence formula_30, its limit is defined by formula_31 where formula_32 is an infinite index. Here the limit is said to exist if the standard part is the same regardless of the infinite index chosen.
Continuity.
A real function formula_25 is continuous at a real point formula_1 if and only if the composition formula_35 is "constant" on the halo of formula_1. See microcontinuity for more details.
References.
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