Closure Properties of Real Number Classes under CBV Functions

Authors:
Xizhong Zheng;Robert Rettinger;Romain Gengler
Affiliations:
Theoretische Informatik, BTU Cottbus, 03044 Cottbus, Germany and Department of Computer Science, Jiangsu University, Zhenjiang, Jiangsu, People's Republic of China;Theoretische Informatik II, FernUniversität Hagen, 58084 Hagen, Germany;Theoretische Informatik, BTU Cottbus, 03044 Cottbus, Germany
Venue:
Theory of Computing Systems
Year:
2005

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Abstract

CBV functions are computable real functions of bounded variation. In this paper we investigate the basic properties of CBV functions. We are especially interested in the question of whether a real number class is closed under CBV functions. The real number classes considered here include the classes of computable (EC), semi-computable (SC), weakly computable (WC), divergence bounded computable (DBC) and recursively approximable (RA) real numbers. We show that the classes EC, RA and DBC are closed under CBV functions but SC and WC are not. Furthermore, WC$ is not even closed under computable monotone functions and, finally, the image sets of $\wc$ under computable monotone functions and CBV functions are different.