Increment theorem

In nonstandard analysis, a field of mathematics, the increment theorem states the following: Suppose a function "f" is differentiable at "x" and that Δ"x" is infinitesimal. Then

:$Delta\; f\; =\; f\text{'}(x),Delta\; x\; +\; varepsilon,\; Delta\; x,$

for some infinitesimal ε, where

: $Delta\; f=f(x+Delta\; x)-f(x).,$

If $scriptstyleDelta\; x\; ot=0$ then we may write

: $frac\{Delta\; f\}\{Delta\; x\}\; =\; f\text{'}(x)+varepsilon,$

which implies that $scriptstylefrac\{Delta\; f\}\{Delta\; x\}approx\; f\text{'}(x)$, or in other words that $scriptstyle\; frac\{Delta\; f\}\{Delta\; x\}$ is infinitely close to $scriptstyle\; f\text{'}(x),$.

See also

*Nonstandard calculus
*Calculus
*Abraham Robinson

References

* H. Jerome Keisler: "Elementary Calculus: An Approach Using Infinitesimals". 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
*cite book|last=Robinson|first=Abraham| title=Non-standard analysis|year=1996| edition=Revised edition | publisher=Princeton University Press| id=ISBN 0-691-04490-2