A Transfer Principle for the Continuations of Real Functions to the Levi-Civita Field

Abstract

We discuss the properties of the continuations of real functions to the Levi-Civita field. In particular, we show that, whenever a function f is analytic on a compact interval [a, b] ⊂ ℝ, f and its analytic continuation f̅_∞ satisfy the same properties that can be expressed in the language of real closed ordered fields. If f is not analytic, then this equivalence does not hold. These results suggest an analogy with the internal and external functions of nonstandard analysis: while the canonical continuations of analytic functions resemble internal functions, the continuations of non-analytic functions behave like external functions. Inspired by this analogy, we suggest some directions for further research.