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On the \(\mathcal{A}\)-continuity of real functions. (English) Zbl 0880.26006
Let \(A\) be a regular matrix. A function \(f:\mathbb{R}\to\mathbb{R}\) is said to be \(A\)-continuous at \(x_0\in\mathbb{R}\) provided that \(A\)-\(\lim x_n=x_0\) implies \(A\)-\(\lim f(x_n)= f(x_0)\) [cf. J. Antoni and T. Šalát, Acta Math. Univ. Comenianae 39, 159-164 (1980; Zbl 0519.40006)]. In connection with the \(A\)-continuity of functions, the authors introduce the concept of \({\mathcal A}\)-continuity of functions replacing the matrix \(A\) by a sequence \({\mathcal A}= (A^i)\) of matrices. In the paper, sufficient conditions are given for usual continuity, linearity and constancy of functions that are \({\mathcal A}\)-continuous at a point \(x_0\in\mathbb{R}\).

MSC:
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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