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The ω-problem

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Kowalczyk S.

Instytut Matematyczny Polskiej Akademii Nauk

Let (X,𝓣) be any T₁ topological space. Given a function F: X → ℝ and x ∈ X, we define the oscillation of F at x to be $ω(F,x) = inf_{U}sup_{x₁,x₂∈ U} |F(x₁) - F(x₂)|$, where the infimum is taken over all neighborhoods U of x. It is well known that ω(F,·): X → [0,∞] is upper semicontinuous and vanishes at all isolated points of X. Suppose an upper semicontinuous function f: X → [0,∞] vanishing at isolated points of X is given. If there exists a function F: X → ℝ such that ω(F,·)=f, then we call F an ω-primitive for f. By the 'ω-problem' on a topological space X we mean the problem of the existence of an ω-primitive for a given upper semicontinuous function vanishing at all isolated points of X. The main topics of the present paper are some results concerning the classical ω-problem and some new generalizations.

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1 Kowalczyk S.

1 2014

Wyniki wyszukiwania

The ω-problem