Non-standard Calculus - Applications

Applications

One immediate application is an extension of the standard definitions of differentiation and integration to internal functions on intervals of hyperreal numbers.

An internal hyperreal-valued function f on is S-differentiable at x, provided

exists and is independent of the infinitesimal h. The value is the S derivative at x.

Theorem. Suppose f is S-differentiable at every point of where b − a is a bounded hyperreal. Suppose furthermore that

Then for some infinitesimal ε

To prove this, let N be a non-standard natural number. Divide the interval into N subintervals by placing N − 1 equally spaced intermediate points:

Then

Now the maximum of any internal set of infinitesimals is infinitesimal. Thus all the ε_k's are dominated by an infinitesimal ε. Therefore,

from which the result follows.