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In partial fulfillment of the requirements for a degree of *Doctor
of Philosophy* from Michigan State University.

After reviewing the algebraic, order, and topological structures of
the field * R* \cite{adtheory,stustibudapest,rscrptsf}, we
review two types of convergence and prove new results about the
convergence of the sums and products of sequences and infinite
series. A weak convergence criterion \cite{stustibudapest} for power
series is then enhanced and proved, and we show that power series can
be reexpanded around any point of their domain of convergence.
Knowledge of weak convergence of power series allows the extension to
the new field and the study of all transcendental functions. This
also will allow the extension of all the real functions that can be
represented on a computer and is thus of great importance for the
implementation of the

We review two different definitions of continuity and
differentiability . We show
that these smoothness criteria are preserved under addition,
multiplication and composition of functions. We show with several
examples that topological continuity and differentiability are not
sufficient to assure that a function be bounded or satisfy any of the
common theorems of real calculus on a closed interval of
* R*. We derive a result which allows for an easy check of
the differentiability of functions. Then, based on the stronger
concept of differentiability, we present a detailed study of a large
class of functions for which we generalize the intermediate value
theorem in and prove an inverse
function theorem.

Based on our knowledge of convergence of power series, we study a
large class of functions which are given locally by power series with
* R* coefficients and which generalize the normal functions
discussed in . We show that the
so-called expandable functions form
an algebra and have all the nice properties of real power series. In
particular, they satisfy the intermediate value theorem, the maximum
theorem and the mean value theorem. Moreover, they are infinitely
often differentiable and integrable; and the derivative functions of
all orders are themselves expandable functions.

The existence of infinitely small numbers in the non-Archimedean field
* R* allows the use of the old numerical
algorithm for computing derivatives of real functions, but now with an
error that in a rigorous way can be shown to become infinitely small (and
hence irrelevant). Using calculus on

K. Shamseddine (1999)

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