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Continuity and derivability of additive interval functions. (English) Zbl 0592.28004

Let \(\phi\) be an additive interval function on a nondegerate closed interval \(I\subset {\mathbb{R}}^ n\). The author establishes conditions which are equivalent to the continuity of \(\phi\) on I. Using these conditions he proves that if \(\phi\) is strongly differentiable (the derivative being finite) at any point of I, then \(\phi\) is continuous on I. It is shown that an analogous assertion is not true for ordinarily differentiable \(\phi\). However, if \(\Psi\) is an additive interval function on an open set \(G\supset I\) which is ordinarily differentiable at any point of I, then \(\Psi\) is continuous on I.

MSC:

28A15 Abstract differentiation theory, differentiation of set functions
26B05 Continuity and differentiation questions
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References:

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