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On the A-continuity of real function. II. (English) Zbl 0615.40002
Two different sufficient conditions are given under which A-continuity in a point implies linearity. The first is property (G): ”a regular matrix summability method A has the property (G) if there exist sequences \((\alpha_ u),(\beta_ u)\in \{0,1\}^{{\mathbb{N}}}\), A-convergent to a,b\(\in]0,1[\), such that \((a/(1-a))^ p\neq (b/(1-b))^ q\), for every p,p nonzero and integer.” The second condition consists in existing of sequences \((\alpha_ u),(\beta_ u),(\gamma_ u)\in \{0,1\}^{{\mathbb{N}}}\) A-convergent to a,b,c\(\in]0,1[\) such that \(\alpha_ u+\beta_ u+\gamma_ u=1\). A-continuity has already been studied by the same author with T. Salat [Acta Math. Univ. Comenianae 39, 159-164 (1980; Zbl 0519.40006)]. The structure of the set of the points of continuity is also studied; in particular one proves that for every \(G_{\delta}\)-set B there exists a regular summability method T and a function f for which \(C_{fT}=B\).
Reviewer: F.Barbieri

MSC:
40C05 Matrix methods for summability
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
Citations:
Zbl 0519.40006
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References:
[1] ANTONI J., ŠALÁT T.: On the A-continuity of real functions. Acta Math. Univ. Comenian. 39, 1980, 159-164. · Zbl 0519.40006
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