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On $$A$$-continuity of real functions. (English) Zbl 0809.26002
The paper is closely related to the paper J. Antoni and the reviewer [Acta Math. Univ. Comen. 39, 159-164 (1980; Zbl 0519.40006)] and it is a continuation of investigations from . 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 . 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 J. Antoni [Math. Slovaca 36, 283-288 (1986; Zbl 0615.40002)]. The paper  is not quoted in the reviewed paper.

MSC:
 26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
Keywords:
$$A$$-continuity
Citations:
Zbl 0519.40006; Zbl 0615.40002