In nonstandard analysis, a field of mathematics, the increment theorem states the following: Suppose a function is differentiable at and that is infinitesimal. Then
for some infinitesimal , 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 is differentiable, but now let be a nonzero standard real number. Then the same equation
holds with the same definition of , but instead of being infinitesimal, we have
(treating and as given so that is a function of alone).
See also Nonstandard calculus Elementary Calculus: An Infinitesimal Approach Abraham Robinson Taylor's theorem
References 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