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 arithmeticoperationswithintervals, suchfunctionsare traditionallytreated by means ofcorresponding 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 analysi...

• Type: Journal
• Year: 2006
• Where: RC
• Authors: Roumen Anguelov, Svetoslav Markov, Blagovest Sendov
• Keywords: H-continuous Functions, Interval Functions, RC 2006, Special Interval-valued Functions