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 , 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.
@book{bwmeta1.element.bwnjournal-rm-doi-10_4064-dm501-0-1,
author = {Stanis\l aw Kowalczyk},
title = {The $\omega$-problem},
series = {GDML\_Books},
year = {2014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm501-0-1}
}
Stanisław Kowalczyk. The ω-problem. GDML_Books (2014), http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm501-0-1/