Basic Theorems
If f is a real valued function defined on an interval, then the transfer operator applied to f, denoted by *f, is an internal, hyperreal-valued function defined on the hyperreal interval .
Theorem. Let f be a real-valued function defined on an interval . Then f is differentiable at a < x < b if and only if for every non-zero infinitesimal h, the value
is independent of h. In that case, the common value is the derivative of f at x.
This fact follows from the transfer principle of non-standard analysis and overspill.
Note that a similar result holds for differentiability at the endpoints a, b provided the sign of the infinitesimal h is suitably restricted.
For the second theorem, we consider the Riemann integral. This integral is defined as the limit, if it exists, of a directed family of Riemann sums; these are sums of the form
where
We will call such a sequence of values a partition or mesh and
the width of the mesh. In the definition of the Riemann integral, the limit of the Riemann sums is taken as the width of the mesh goes to 0.
Theorem. Let f be a real-valued function defined on an interval . Then f is Riemann-integrable on if and only if for every internal mesh of infinitesimal width, the quantity
is independent of the mesh. In this case, the common value is the Riemann integral of f over .
Read more about this topic: Non-standard Calculus
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