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
.