## 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)].

### MSC:

 65G40 General methods in interval analysis

Zbl 1046.15002
Full Text:

### References:

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