Editing Nonstandard calculus (section)

==Continuity==
A real function ''f'' is continuous at a standard real number ''x'' if for every hyperreal ''x' '' infinitely close to ''x'', the value ''f''(''x' '') is also infinitely close to ''f''(''x'').  This captures [[Cauchy]]'s definition of continuity as presented in his 1821 textbook [[Cours d'Analyse]], p.&nbsp;34.

Here to be precise, ''f'' would have to be replaced by its natural hyperreal extension usually denoted ''f''<sup>*</sup>.

Using the notation <math>\approx</math> for the relation of being infinitely close as above,
the definition can be extended to arbitrary (standard or nonstandard) points as follows:

A function ''f'' is ''[[microcontinuous]]'' at ''x'' if whenever <math>x'\approx x</math>, one has <math>f^*(x')\approx f^*(x)</math>

Here the point x' is assumed to be in the domain of (the natural extension of) ''f''.

The above requires fewer quantifiers than the [[(ε, δ)-definition of limit|(''ε'',&nbsp;''δ'')-definition]] familiar from  standard elementary calculus:

''f'' is continuous at ''x'' if for every ''ε''&nbsp;>&nbsp;0, there exists a ''δ''&nbsp;>&nbsp;0 such that for every ''x' '', whenever |''x''&nbsp;&minus;&nbsp;''x'&thinsp;''|&nbsp;<&nbsp;''δ'', one has |''f''(''x'')&nbsp;&minus;&nbsp;''f''(''x'&thinsp;'')|&nbsp;<&nbsp;''ε''.