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

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)


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


Zbl 0940.26003
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