Increment theorem
Encyclopedia
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 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
.
Non-standard analysis
Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...
, a field of mathematics, the increment theorem states the following: Suppose a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
y = f(x) is differentiable at x and that Δx is infinitesimal
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
. 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 partStandard part function
In non-standard analysis, the standard part function is a function from the limited hyperreals to the reals, which associates to every hyperreal, the unique real infinitely close to it. As such, it is a mathematical implementation of the historical concept of adequality introduced by Pierre de...
of
.See also
- Non-standard calculusNon-standard calculusIn mathematics, non-standard calculus is the modern application of infinitesimals, in the sense of non-standard analysis, to differential and integral calculus...
- Elementary Calculus: An Infinitesimal ApproachElementary Calculus: An Infinitesimal ApproachElementary Calculus: An Infinitesimal approach is a textbook by H. Jerome Keisler. The subtitle alludes to the infinitesimal numbers of Abraham Robinson's non-standard analysis...
- Abraham RobinsonAbraham RobinsonAbraham Robinson was a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics....