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Smallness of sets of nondifferentiability of convex functions in non- separable Banach spaces. (English) Zbl 0768.58005

Some results concerning the smallness of sets of nondifferentiability (in the sense of Fréchet and Gâteaux) of continuous real functions defined on non-separable Banach spaces are obtained. In fact the theorems are directly formulated in terms of multivalued monotone operators.
The author uses two types of smallness for the sets in a Banach space: cone-small sets and \(\sigma\)-cone supported sets. Both are \(\sigma\)- porous.
For example the following theorem is proved: if \(T\) is a maximal monotone operator on an Asplund space and having a domain \(D(T)\) with a nonempty interior \(G\), then there exist a \(\sigma\)-cone supported set \(A\) and a cone-small set \(B\) contained in \(G\) such that \(T\) is single-valued and norm-to-norm upper semicontinuous at each point of \(G\setminus (A\cup B)\).

MSC:

58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
47H05 Monotone operators and generalizations
46G05 Derivatives of functions in infinite-dimensional spaces
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References:

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