Abstract
The formalization of correctness and optimality concepts, which are desirable requirements for interval (and other self-validated methods) libraries, is related to topological aspects of real numbers and Moore arithmetics, characterizing some families of interval functions. The main motivation is that real continuous functions are largely used in many fields of human activity, and a characterization of their interval representation in terms of interval topologies leads to the knowledge of how interval algorithms are suitable to represent those functions. In this paper, we characterize and relate some classes of interval functions with respect to three natures of an interval (as a set, as an information, as a number). These natures establish different ways to classify intervals, and hence different notions of continuity. Here we relate the notion of interval representations with those classes of functions.
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Notes
Observe that \(d(a,b)=d(b,a)=0\) and \(a\ne b\) do not hold, but \(d(a,b)=d(b,a)\ne 0\) and \(a=b\), and \(d(a,b)=d(b,a)\ne 0\) and \(a\ne b\) can hold.
Note that when \(q\) is a metric, \(q,\overline{q},\) and \(q^*\) coincide and the topological spaces are the same.
More formally \(F\) is bi-continuous if for every bi-directed set—let \(\mathcal D =(D,\le )\) be a poset. \(\Delta \subseteq D\) is bi-directed if \(\Delta \) is a directed set of \(\mathcal D \) and \(\mathcal D ^\mathrm{op}=(D,\ge )\)—\(F(\bigsqcup \Delta )=\bigsqcup F(\Delta )\) and
.For us, a real function \(f\) is asymptotic if for some interval \([a,b]\), the set \(\{f(x^{\prime }): a\le x^{\prime }\le b\}\) has no supremum or no infimum.
For every interval \(A\), \(F(A)\sqsubseteq G(A)\).
For every interval \(A\), \(F(A)\sqsubseteq \mathrm{CIR}(f)(A)\).
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Acknowledgments
This work was partially supported by Brazilian Research Council (CNPq) under the processes numbers 306876/2012-4, 307681/2012-2 and 480832/2011-0.
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Communicated by Renata Hax Reiser Sander.
The partial results described in this paper were previously presented in IntMath TDS at CNMAC 2010-SBMAC honoring Prof. Dalcidio Claudio and his leadership over more than 40 years of Interval Mathematics research in Brazil.
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Bedregal, B., Santiago, R. Some continuity notions for interval functions and representation. Comp. Appl. Math. 32, 435–446 (2013). https://doi.org/10.1007/s40314-013-0049-z
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DOI: https://doi.org/10.1007/s40314-013-0049-z