# Increment theorem

In non-standard analysis, a field of mathematics, the **increment theorem** states the following: Suppose a function *y* = *f*(*x*) is differentiable at *x* and that Δ*x* is infinitesimal. Then

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 Δ*x* be a nonzero standard real number. Then the same equation

holds with the same definition of Δ*y*, but instead of ε being infinitesimal, we have

(treating *x* and *f* as given so that ε is a function of Δ*x* alone).

## 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 - Robinson, Abraham (1996).
*Non-standard analysis*(Revised ed.). Princeton University Press. ISBN 0-691-04490-2.

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