The ω-problem

Stanisław Kowalczyk. The ω-problem. 2014. <http://eudml.org/doc/286063>.

@book{StanisławKowalczyk2014,
abstract = {Let (X,) 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.},
author = {Stanisław Kowalczyk},
language = {eng},
title = {The ω-problem},
url = {http://eudml.org/doc/286063},
year = {2014},
}

TY - BOOK
AU - Stanisław Kowalczyk
TI - The ω-problem
PY - 2014
AB - Let (X,) 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.
LA - eng
UR - http://eudml.org/doc/286063
ER -