One-Variable and Multi-Mariable Calculus on a Non-Archimedean Field Extension of the Real Numbers∗

New elements of calculus on a complete real closed non-Archimedean field extension F of the real numbers R will be presented. It is known that the total disconnectedness of F in the topology induced by the order makes the usual (topological) notions of continuity and differentiability too weak to extend real calculus results to F . In this paper, we introduce new stronger concepts of continuity and differentiability which we call derivate continuity and derivate differentiability [2, 12]; and we show that derivate continuous and differentiable functions satisfy the usual addition, product and composition rules and that n-times derivate differentiable functions satisfy a Taylor formula with remainder similar to that of the real case. Then we generalize the definitions of derivate continuity and derivate differentiability to multivariable F-valued functions and we prove related results that are useful for doing analysis on F and F in general. DOI: 10.1134/S2070046613020040