International Journal of Mathematics and Mathematical Sciences
Volume 30 (2002), Issue 3, Pages 165-176

Intermediate values and inverse functions on non-Archimedean fields

Khodr Shamseddine1 and Martin Berz2

1Department of Mathematics, Michigan State University, East Lansing 48824, MI, USA
2Department of Physics and Astronomy, Michigan State University, East Lansing 48824, MI, USA

Received 27 April 2001; Revised 20 September 2001

Copyright © 2002 Khodr Shamseddine and Martin Berz. 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.


Continuity or even differentiability of a function on a closed interval of a non-Archimedean field are not sufficient for the function to assume all the intermediate values, a maximum, a minimum, or a unique primitive function on the interval. These problems are due to the total disconnectedness of the field in the order topology. In this paper, we show that differentiability (in the topological sense), together with some additional mild conditions, is indeed sufficient to guarantee that the function assumes all intermediate values and has a differentiable inverse function.