Editing Continuous function (section)

====Definition using the hyperreals====
[[Cauchy]] defined the continuity of a function in the following intuitive terms: an [[infinitesimal]] change in the independent variable corresponds to an infinitesimal change of the dependent variable (see ''Cours d'analyse'', page 34). [[Non-standard analysis]] is a way of making this mathematically rigorous. The real line is augmented by adding infinite and infinitesimal numbers to form the [[hyperreal numbers]]. In nonstandard analysis, continuity can be defined as follows.

{{block indent|em=1.5|text=A real-valued function {{math|''f''}} is continuous at {{mvar|x}} if its natural extension to the hyperreals has the property that for all infinitesimal {{math|''dx''}}, <math>f(x + dx) - f(x)</math> is infinitesimal<ref>{{cite web| url=http://www.math.wisc.edu/~keisler/calc.html |title=Elementary Calculus|work=wisc.edu}}</ref>}}

(see [[microcontinuity]]).  In other words, an infinitesimal increment of the independent variable always produces an infinitesimal change of the dependent variable, giving a modern expression to [[Augustin-Louis Cauchy]]'s definition of continuity.