Abstract
In this paper, we review the definition and properties of locally uniformly differentiable functions on N, a non-Archimedean field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. Then we define and study n-times locally uniform differentiable functions at a point or on a subset of N. In particular, we study the properties of twice locally uniformly differentiable functions and we formulate and prove a local mean value theorem for such functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Berz, “Calculus and numerics on Levi-Civita fields,” in Computational Differentiation: Techniques, Applications, and Tools, M. Berz, C. Bischof, G. Corliss and A. Griewank, editors, 19–35 (SIAM, Philadelphia, 1996).
H. Hahn, “Über die nichtarchimedischen Größensysteme,” Sitzungsbericht der Wiener Akademie der Wissenschaften Abt. 2a 117, 601–655 (1907).
W. Krull, “Allgemeine Bewertungstheorie,” J. Reine Angew.Math. 167, 160–196 (1932).
T. Levi-Civita, “Sugli infiniti ed infinitesimi attuali quali elementi analitici,” Atti Ist. Veneto di Sc., Lett. ed Art. 7a, 4, 1765 (1892).
T. Levi-Civita, “Sui numeri transfiniti,” Rend. Acc. Lincei 5a, 7 (91), 113 (1898).
S. MacLane, “The universality of formal power series fields,” Bull. Amer.Math. Soc. 45, 888 (1939).
S. Priess-Crampe, Angeordnete Strukturen: Gruppen, Körper, projektive Ebenen (Springer, Berlin, 1983).
F. J. Rayner, “Algebraically closed fields analogous to fields of Puiseux series,” J. London Math. Soc. 8, 504–506 (1974).
P. Ribenboim, “Fields: Algebraically closed and others,” Manuscr. Math. 75, 115–150 (1992).
A. Robinson, Non-Standard Analysis (North-Holland, 1974).
W. H. Schikhof, Ultrametric Calculus: An Introduction to p-Adic Analysis (Cambridge Univ. Press, 1985).
Ya. D. Sergeyev, “Numerical point of view on calculus for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains,” Nonlin. Anal. Ser. A: Theo.Meth. Appl. 71 (12), 1688–1707 (2009).
Ya. D. Sergeyev, “Higher order numerical differentiation on the infinity computer,” Optim. Lett. 5 (4), 575–585 (2011).
Ya. D. Sergeyev, “Solving ordinary differential equations by working with infinitesimals numerically on the infinity computer,” Appl.Math. Comput. 219 (22), 10668–10681 (2013).
K. Shamseddine, “Analysis on the Levi-Civita field and computational applications,” J. Appl. Math. Comp. 255, 44–57 (2015).
K. Shamseddine and M. Berz, “Exception handling in derivative computation with non-Archimedean calculus,” in Computational Differentiation: Techniques, Applications and Tools, M. Berz, C. Bischof, G. Corliss and A. Griewank, editors, 37–51 (SIAM, Philadelphia, 1996).
K. Shamseddine and M. Berz, “Analysis on the Levi-Civita field, a brief overview,” Contemp. Math. 508, 215–237 (2010).
K. Shamseddine, T. Rempel and T. Sierens, “The implicit function theorem in a non-Archimedean setting,” Indag. Math. (N.S.) 20(4), 603–617 (2009).
K. Shamseddine and T. Sierens, “On locally uniformly differentiable functions on a complete non-Archimedean ordered field extension of the real numbers,” ISRN Math. Anal. 2012, 20 pages (2012).
N. Vakil, Real Analysis through Modern Infinitesimals (Cambridge Univ. Press, 2011).
Author information
Authors and Affiliations
Corresponding author
Additional information
The text was submitted by the authors in English.
Rights and permissions
About this article
Cite this article
Shamseddine, K., Bookatz, G. A local mean value theorem for functions on non-archimedean field extensions of the real numbers. P-Adic Num Ultrametr Anal Appl 8, 160–175 (2016). https://doi.org/10.1134/S2070046616020059
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S2070046616020059