In nonstandard analysis, a field of mathematics, the increment theorem states the following: Suppose a function y = f(x) is differentiable at x and that Δx 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 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).

See also