# The Set of Hausdorff Continuous Functions— The Largest Linear Space of Interval Functions

The Set of Hausdorff Continuous Functions— The Largest Linear Space of Interval Functions Hausdorff continuous (H-continuous) functions are special interval-valued functions which are commonly used in practice, e.g. histograms are such functions. However, in order to avoid arithmetic operations with intervals, such functions are traditionally treated by means of corresponding semi-continuous functions, which are real-valued functions. One difficulty in using H-continuous functions is that, if we add two H-continuous functions that have interval values at same argument using point-wise interval arithmetic, then we may obtain as a result an interval function which is not H-continuous. In this work we define addition so that the set of H-continuous functions is closed under this operation. Moreover, the set of H-continuous functions is turned into a linear space. It has been also proved that this space is the largest linear space of interval functions. These results make H-continuous functions an attractive tool in real analysis and provides a bridge between real and interval analysis. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Reliable Computing Springer Journals

# The Set of Hausdorff Continuous Functions— The Largest Linear Space of Interval Functions

, Volume 12 (5) – Jul 29, 2006
27 pages

/lp/springer_journal/the-set-of-hausdorff-continuous-functions-the-largest-linear-space-of-ebAYAlRP8e
Publisher
Springer Journals
Subject
Mathematics; Numeric Computing; Mathematical Modeling and Industrial Mathematics; Approximations and Expansions; Computational Mathematics and Numerical Analysis
ISSN
1385-3139
eISSN
1573-1340
D.O.I.
10.1007/s11155-006-9006-5
Publisher site
See Article on Publisher Site

### Abstract

Hausdorff continuous (H-continuous) functions are special interval-valued functions which are commonly used in practice, e.g. histograms are such functions. However, in order to avoid arithmetic operations with intervals, such functions are traditionally treated by means of corresponding semi-continuous functions, which are real-valued functions. One difficulty in using H-continuous functions is that, if we add two H-continuous functions that have interval values at same argument using point-wise interval arithmetic, then we may obtain as a result an interval function which is not H-continuous. In this work we define addition so that the set of H-continuous functions is closed under this operation. Moreover, the set of H-continuous functions is turned into a linear space. It has been also proved that this space is the largest linear space of interval functions. These results make H-continuous functions an attractive tool in real analysis and provides a bridge between real and interval analysis.

### Journal

Reliable ComputingSpringer Journals

Published: Jul 29, 2006

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