Increment theorem

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
• '
• Abraham Robinson
• Taylor's theorem

• Howard Jerome Keisler: '. 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