Editing Nonstandard calculus (section)

== Basic theorems ==

If ''f'' is a real valued function defined on an interval [''a'', ''b''], then the transfer operator applied to ''f'', denoted by ''*f'',  is an ''internal'', hyperreal-valued function defined on the hyperreal interval [*''a'', *''b''].

''Theorem'':  Let ''f'' be a real-valued function defined on an
interval [''a'', ''b'']. Then ''f'' is differentiable at ''a < x < b'' if and only if
for every ''non-zero'' infinitesimal ''h'', the value

:<math> \Delta_h f := \operatorname{st} \frac{[{}^*\!f](x+h)-[{}^*\!f](x)}{h} </math>

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 nonstandard 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, the Riemann integral is defined as the limit, if it exists, of a directed family of ''Riemann sums''; these are sums of the form

:<math> \sum_{k=0}^{n-1} f(\xi_k) (x_{k+1} - x_k) </math>

where

:<math>a = x_0 \leq \xi_0 \leq x_1 \leq \ldots x_{n-1} \leq \xi_{n-1} \leq x_n = b.</math>

Such a sequence of values is called a ''partition'' or ''mesh'' and

:<math> \sup_k (x_{k+1} - x_k) </math>

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 [''a'', ''b'']. Then ''f'' is Riemann-integrable on [''a'', ''b''] if and only if
for every internal mesh of infinitesimal width, the quantity

:<math> S_M = \operatorname{st} \sum_{k=0}^{n-1} [*f](\xi_k) (x_{k+1} - x_k) </math>

is independent of the mesh.  In this case, the common value is
the Riemann integral of ''f'' over [''a'', ''b''].