On the sets of discontinuity points of functions satisfying some approximate quasi-continuity conditions. (English) Zbl 1057.26005

Grande, Marcin

Real Anal. Exch. 27(2001-2002), No. 2, 773-781 (2002).

In [Z. Grande, Real Anal. Exch. 24, No.1, 171-183 (1998; Zbl 0940.26003)], there is defined when a function \(f:\mathbb R\to\mathbb R\) satisfies the condition (\(s_i\)), \(i=0,1,2,3\). In the present paper, the author characterizes the family \(X(S_i)\) of sets \(A\) for which there exists a function \(f\) satisfying \(s_i\) such that \(A\) is the set of all discontinuity points of \(f\). He also proves that \(X(S_1)=X(S_3)\), \(X(S_0)=X(S_2)\) and that for each \(A\in X(S_1)\) there exists an \((s_1)\) function \(f\) such that \(A\) is the set of all approximate discontinuity points of \(f\).

Reviewer: Jan Malý (Praha)

MSC:

26A15	Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
28A20	Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
54C08	Weak and generalized continuity
26A03	Foundations: limits and generalizations, elementary topology of the line

Keywords:

density topology; approximate continuity; generalized continuity; discontinuity points

Citations:

Zbl 0940.26003