The set of Hausdorff continuous functions – the largest linear space of interval functions. (English) Zbl 1110.65036

Let \(\mathbb{R}\) denote the set of real compact intervals \([a]= [\underline a, \overline a]\) with \(|[a]|=\max\{|\underline a|,|\overline a|\}\), and let \(\Omega\) be an open subset of \(\mathbb{R}^n\). A function \(f: \Omega\to\mathbb{I}\mathbb{R}\), is called \(f\)-locally bounded if for every \(x\in \Omega\) there exist \(\delta>0\) and \(M\in\mathbb{R}\) such that \(|f(y)|<M\) for all \(y\in B_\delta(x):=\{y\in\Omega|\|x-y\| <\delta\}\), \(\|\cdot\|\) being some norm in \(\mathbb{R}^n\). Denote by \(\mathbb{A}(\Omega)\) the set of all such \(f\)-locally bounded functions and define for a dense subset \(D\) of \(\Omega\) the mapping \(F:\mathbb{A}(D)\to\mathbb{A}(\Omega)\) by
\[ F(D, \Omega,f)(x): =\left[\sup_{\delta>0}\inf\{f(y)\mid y\in B_\delta(x)\cap D\},\;\inf_{\delta>0} \sup\bigl\{f(y)\mid y\in B_\delta(x)\cap D\bigr\} \right]. \]
Let \(f\in\mathbb{A} (\Omega)\). If \(f\) satisfies \(F(\Omega,\Omega, f)=f\) it is called \(S\)-continuous. If for every dense subset \(D\) of \(\Omega\) the property \(F(D,\Omega,f)=f\) is fulfilled then \(f\) is called \(D\)-continuous. If \(F(\Omega,\Omega,g)(x)=f(x)\) holds for every function \(g\in\mathbb{A}(\Omega)\) with \(g(x)\subset f(x)\), \(x\in \Omega\), then \(f\) is called Hausdorff-continuous. The authors present properties of \(S\)-, \(D\)- and \(H\)-continuous functions, among them relations to semi-continuity of real functions, to continuity and to the interval hull of continuous functions. Moreover, they introduce operations \(\oplus,*\) in the set \(\mathbb{H}(\Omega)\) of \(H\)-continuous functions such that \(\mathbb{H} (\Omega)\) becomes a linear space over the field \(\mathbb{R}\). It is proved that in some sense this space is the largest linear space within the set \(\mathbb{G}(\Omega)\) of \(D\)-continuous functions defined on \(\Omega\). Extending the operations from \(\mathbb{H}(\Omega)\) to \(\mathbb{G}(\Omega)\) in an appropriate way shows that \(\mathbb{G} (\Omega)\) is a quasi-linear space as defined by [S. Markov, J. Comput. Appl. Math. 162, No. 1, 93–112 (2004; Zbl 1046.15002)].


65G40 General methods in interval analysis


Zbl 1046.15002
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