for some infinitesimal ε, where
If then we may write
which implies that , or in other words that is infinitely close to , or is the standard part of .
A similar theorem exists in standard Calculus. Again assume that y = f(x) is differentiable, but now let Δx be a nonzero standard real number. Then the same equation
holds with the same definition of Δy, but instead of ε being infinitesimal, we have
(treating x and f as given so that ε is a function of Δx alone).
- Howard Jerome Keisler: Elementary Calculus: An Infinitesimal Approach. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html
- Robinson, Abraham (1996). Non-standard analysis (Revised ed.). Princeton University Press. ISBN 0-691-04490-2.