Editing Nonstandard analysis (section)

== First consequences ==
{{see also|microcontinuity}}

The symbol {{math|*'''N'''}} denotes the nonstandard natural numbers. By the extension principle, this is a superset of {{math|'''N'''}}. The set {{math|*'''N''' − '''N'''}} is nonempty. To see this, apply countable [[saturated model|saturation]] to the sequence of internal sets

:<math> A_n = \{k \in {^*\mathbf{N}}: k \geq n\} </math>

The sequence {{math|{''A<sub>n</sub>''}<sub>''n'' ∈ '''N'''</sub>}} has a nonempty intersection, proving the result.

We begin with some definitions: Hyperreals ''r'', ''s'' are ''infinitely close'' [[if and only if]]

:<math> r \cong s \iff \forall \theta \in \mathbf{R}^+, \ |r - s| \leq \theta</math>

A hyperreal {{mvar|r}} is ''infinitesimal'' if and only if it is infinitely close to 0. For example, if {{mvar|n}} is a [[hyperinteger]], i.e. an element of {{math|*'''N''' − '''N'''}}, then {{math|1/''n''}} is an infinitesimal. A hyperreal {{mvar|r}} is ''limited'' (or ''finite'') if and only if its absolute value is dominated by (less than) a standard integer. The limited hyperreals form a subring of {{math|*'''R'''}} containing the reals. In this ring, the infinitesimal hyperreals are an [[ideal (ring theory)|ideal]].

The set of limited hyperreals or the set of infinitesimal hyperreals are ''external'' subsets of {{math|''V''(*'''R''')}}; what this means in practice is that bounded quantification, where the bound is an internal set, never ranges over these sets.

'''Example''': The plane {{math|(''x'', ''y'')}} with {{mvar|x}} and {{mvar|y}} ranging over {{math|*'''R'''}} is internal, and is a model of plane Euclidean geometry. The plane with {{mvar|x}} and {{mvar|y}} restricted to limited values (analogous to the [[Dehn planes|Dehn plane]]) is external, and in this limited plane the parallel postulate is violated. For example, any line passing through the point {{math|(0, 1)}} on the {{mvar|y}}-axis and having infinitesimal slope is parallel to the {{mvar|x}}-axis.

'''Theorem.''' For any limited hyperreal {{mvar|r}} there is a unique standard real denoted {{math|st(''r'')}} infinitely close to {{mvar|r}}. The mapping {{math|st}} is a ring homomorphism from the ring of limited hyperreals to {{math|'''R'''}}.

The mapping st is also external.

One way of thinking of the [[standard part function|standard part]] of a hyperreal, is in terms of [[Dedekind cut]]s; any limited hyperreal {{mvar|s}} defines a cut by considering the pair of sets {{math|(''L'', ''U'')}} where {{mvar|L}} is the set of standard rationals {{mvar|a}} less than {{mvar|s}} and {{mvar|U}} is the set of standard rationals {{mvar|b}} greater than {{mvar|s}}. The real number corresponding to {{math|(''L'', ''U'')}} can be seen to satisfy the condition of being the standard part of {{mvar|s}}.

One intuitive characterization of continuity is as follows:

'''Theorem.''' A real-valued function {{mvar|f}} on the interval {{math|[''a'', ''b'']}} is continuous if and only if for every hyperreal {{mvar|x}} in the interval {{math|*[''a'', ''b'']}}, we have: {{math|*''f''(''x'') ≅ *''f''(st(''x''))}}.

Similarly,

'''Theorem.''' A real-valued function {{mvar|f}} is differentiable at the real value {{mvar|x}} if and only if for every infinitesimal hyperreal number {{mvar|h}}, the value

:<math> f'(x)= \operatorname{st} \left(\frac{{^*f}(x+h) - {^*f}(x)}{h}\right) </math>

exists and is independent of {{mvar|h}}. In this case {{math|''f''′(''x'')}} is a real number and is the derivative of {{mvar|f}} at {{mvar|x}}.