Separately continuous functions: approximations, extensions, and restrictions

Lebesgue Measurability of Separately Continuous Functions and Separability

A connection between the separability and the countable chain condition of spaces with L-property (a topological space X has L-property if for every topological space Y , separately continuous function f : X ×Y →R and open set I ⊆R, the set f −1(I) is an Fσ-set) is studied. We show that every completely regular Baire space with the L-property and the countable chain condition is separable and c...

Hausdorff Continuous Interval Functions and Approximations

The set of interval Hausdorff continuous functions constitutes the largest space preserving basic algebraic and topological structural properties of continuous functions, such as linearity, ring structure, Dedekind order completeness, etc. Spaces of interval functions have important applications not only in the construction of numerical methods and algorithms, but to problems in abstract areas ...

Canonical utility functions and continuous preference extensions

I define canonical utility functions via an explicit formula that inherits semicontinuity, continuity, Cauchy continuity, and uniform continuity from preferences. This construction is used to (i) show Rader’s and Debreu’s theorems in a fast and transparent way, (ii) refine these results for Cauchy and uniformly continuous preferences on a metric space X , (iii) extend such preferences from X to...

Self-adjoint Extensions of Restrictions

We provide, by a resolvent Krĕın-like formula, all selfadjoint extensions of the symmetric operator S obtained by restricting the self-adjoint operator A : D(A) ⊆ H → H to the dense, closed with respect to the graph norm, subspace N ⊂ D(A). Neither the knowledge of S∗ nor of the deficiency spaces of S is required. Typically A is a differential operator and N is the kernel of some trace (restric...

Triangular approximations of fuzzy number with value and ambiguity functions