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==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 <math>\mathbb{R}\subset {}^*\mathbb{R}</math> which is not internal; in fact every internal set in <math>{}^\ast\mathbb{R}</math> which is a subset of <math>\mathbb{R}</math> is necessarily ''finite'', see (Goldblatt, 1998).
==Applications==
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
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