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On \(A\)-continuity of real functions. (English) Zbl 0809.26002
The paper is closely related to the paper [1]J. Antoni and the reviewer [Acta Math. Univ. Comen. 39, 159-164 (1980; Zbl 0519.40006)] and it is a continuation of investigations from [1]. Let \(A\) be a regular matrix. A function \(f: \mathbb{R}\to \mathbb{R}\) is said to be \(A\)-continuous at \(x_ 0\in \mathbb{R}\) if \(A\)-\(\lim x_ n= x_ 0\) implies \(A\)-\(\lim f(x_ n)= x_ 0\) for an arbitrary sequence \((x_ n)^ \infty_{n=1}\) of real numbers. The authors prove four theorems. Three of them give sufficient conditions for linearity of a function which is \(A\)-continuous at one or three points of \(\mathbb{R}\). The fourth theorem concerns a special matrix \(B\) investigated in [1]. Finally, the authors formulate the following interesting hypothesis: Let \(A\) be a regular matrix. If there exists a nonlinear function \(g: \mathbb{R}\to \mathbb{R}\) which is \(A\)-continuous at each \(x\in \mathbb{R}\), then every continuous function \(f: \mathbb{R}\to \mathbb{R}\) is \(A\)-continuous at each \(x\in \mathbb{R}\).
Reviewer’s remark: Theorem 2 of this paper coincides with Theorem 1 of the paper [2]J. Antoni [Math. Slovaca 36, 283-288 (1986; Zbl 0615.40002)]. The paper [2] is not quoted in the reviewed paper.

26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable