Intermediate values and inverse functions on non-Archimedean fields

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...

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Main Authors: Khodr Shamseddine, Martin Berz
Format: Article
Language:English
Published: Wiley 2002-01-01
Series:International Journal of Mathematics and Mathematical Sciences
Online Access:http://dx.doi.org/10.1155/S0161171202013030
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author Khodr Shamseddine
Martin Berz
author_facet Khodr Shamseddine
Martin Berz
author_sort Khodr Shamseddine
collection DOAJ
description 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.
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spelling doaj-art-2edbb112506549ada46bf689b5987fa72025-02-03T01:09:48ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252002-01-0130316517610.1155/S0161171202013030Intermediate values and inverse functions on non-Archimedean fieldsKhodr Shamseddine0Martin Berz1Department of Mathematics, Michigan State University, East Lansing, MI 48824, USADepartment of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USAContinuity 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.http://dx.doi.org/10.1155/S0161171202013030
spellingShingle Khodr Shamseddine
Martin Berz
Intermediate values and inverse functions on non-Archimedean fields
International Journal of Mathematics and Mathematical Sciences
title Intermediate values and inverse functions on non-Archimedean fields
title_full Intermediate values and inverse functions on non-Archimedean fields
title_fullStr Intermediate values and inverse functions on non-Archimedean fields
title_full_unstemmed Intermediate values and inverse functions on non-Archimedean fields
title_short Intermediate values and inverse functions on non-Archimedean fields
title_sort intermediate values and inverse functions on non archimedean fields
url http://dx.doi.org/10.1155/S0161171202013030
work_keys_str_mv AT khodrshamseddine intermediatevaluesandinversefunctionsonnonarchimedeanfields
AT martinberz intermediatevaluesandinversefunctionsonnonarchimedeanfields