Hausdorff Continuous Interval Functions and Approximations
The set of interval Hausdorff continuous functions constitutes the largest space preserving basic algebraic and topological structural properties of continuous functions, such as linearity, ring structure, Dedekind order completeness, etc. Spaces of interval functions have important applications not only in the construction of numerical methods and algorithms, but to problems in abstract areas such as real analysis, set-valued analysis, approximation theory and the analysis of PDEs. In this work, we summarize some basic results about the family of interval Hausdorff continuous functions that make interval analysis a bridge between numerical and real analysis. We focus on some approximation issues formulating a new result on the Hausdorff approximation of Hausdorff continuous functions by interval step functions. The Hausdorff approximation of the Heaviside interval step function by sigmoid functions arising from biological applications is also considered, and an estimate for the Hausdorff distance is obtained.
KeywordsInterval functions Baire semi-continuous functions Hausdorff continuous functions Dilworth continuous functions Sigmoid functions
RA acknowledges partial support of the National Research Foundation of South Africa. RA and SM acknowledge partial support by the Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences. The authors thank Prof. Kamen Ivanov for the analysis and derivation of formula (9). They are grateful to the anonimous reviewer for his careful reading and many remarks.
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