Table of Contents
ISRN Mathematical Analysis
Volume 2012, Article ID 387053, 20 pages
Research Article

On Locally Uniformly Differentiable Functions on a Complete Non-Archimedean Ordered Field Extension of the Real Numbers

1Department of Physics and Astronomy, University of Manitoba, Winnipeg, MB, Canada R3T 2N2
2Department of Physics and Astronomy, University of Waterloo, Waterloo, ON, Canada N2L 3G1

Received 20 December 2011; Accepted 24 January 2012

Academic Editors: J.-F. Colombeau and G. Mantica

Copyright © 2012 Khodr Shamseddine and Todd Sierens. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study the 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. In particular, we show that locally uniformly differentiable functions are 𝐶1, they include all polynomial functions, and they are closed under addition, multiplication, and composition. Then we formulate and prove a version of the inverse function theorem as well as a local intermediate value theorem for these functions.