Algebraic Computations with Hausdorff Continuous Functions

Roumen Anguelov and Svetoslav Markov

University of Pretoria, South Africa
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences

CAS typically deal with computations, operations and manipulations of functions of the set of what is considered elementary functions. These functions are all analytic on their domains of definition. To the authors' best knowledge discontinuous functions have not yet been considered in this context. In this paper we consider algebraic computations with Hausdorff continuous (H-continuous) interval valued functions. The H-continuous functions are useful in representing discontinuities of real functions through interval values. They were originally defined within the realm of the Approximation Theory but have been applied since in many other areas. In particular, recent results have shown that they are also a powerful tool in the Analysis of PDEs since the solution of large classes of nonlinear PDEs can be assimilated with H-continuous functions, [3]. It was also shown recently, see [2], that the algebraic operations for continuous functions can be extended to H-continuous functions in such a way that the set of H-continuous functions is a commutative ring. This result is particularly significant in view of the fact that the interval structures typically do not form linear spaces. In fact, the set of H-continuous functions is the largest set of interval functions which is a linear space, [1]. The present paper deals with the automation of the algebraic operations with H-continuous functions within the structure of a functoid.

[1] Anguelov, R., Markov, S., Sendov, B.: The Set of Hausdorff Continuous Functions -- the Largest Linear Space of Interval Functions, Reliable Computing 12, 2006, 337-363.

[2] Anguelov, R., Markov, S., Sendov, B.: Algebraic operations for H-continuous functions, Proc. Internat. Conf. on Constructive Theory of Functions, Varna, to appear.

[3] Anguelov, R., Rosinger,E. E., Hausdorff Continuous Solutions of Nonlinear PDEs through the Order Completion Method, Quaestiones Mathematicae 28(3), 2005, 271-285.