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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
Zbl 0519.40006
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##### References:
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