A Brief Introduction to Infinitesimal Calculus
Section 2: Keisler's Axioms
The following presentation of Keisler's foundations for Robinson's Theory of Infinitesimals is explained in more detail in either of the (free .pdf) files: on my web site and the Epilog to Keisler's text, on his web site.
Small, Medium, and Large Hyperreal Numbers
Keisler's Function Extension Axiom
Continuity & Extreme Values
Microscopic tangency in one variable
The Fundamental Theorem of Integral Calculus
The Local Inverse Function Theorem
Second Differences and Higher Order Smoothness
Created by Mathematica (September 22, 2004)