On the $A$-continuity of real function. II.
Jozef Antoni · Czech digital mathematics library · 1986
In the present paper two problems concerning the A-continuity to a regular matrix summability method are partially solved. Let A = (a™) denote a regular summability method given by a matrix (amn). We A say that a real function / is A-continuous at the point x0 if f(xn)—>f(x0) whenever A x n —> x 0. R. C. Buck [2] showed that if / is a (C, l)-continuous at least at one point of R, then / is a linear function. In paper [1] the existence of a regular matrix summability method A for which there exists a nonlinear function A-continuous at least at one point is given. Professor Salat puts the following problem: 1. To characterize regular summability methods A for which there exists a nonlinear function which is A-continuous at least at one poit. 2. To characterize QA, the set of all points of A-continuity of the function /. method is given for which only linear functions are A-continuous at least at one point. Definition 1. A regular matrix summability method has the property (G) if there exists sequences {a„}*-,l5 {j3„}*=i, of zeros and ones which are A-covergent to numbers a, b respectively a6(0,1), b=£0, b+\\, ( j =£ ( , J for all non-zero integers p, q. Lemma 1. Let T be a regular matrix summability method which sums at least one sequence of zeros and ones to a number a, ai = 0, aj = 1. Letf be a T-continuos at least at one point. Then f is a continuous function. Proof. Let / be a T-continuos at a point z0. Let us suppose that / is discontinuos at a point x. Thus there exists a sequence u„—>0 such that lim f(x + u„) = y^f(x) (also be y-h oo, or — oo). Let {an}n ^ denote a sequence of zeros and ones for which T-lim an = a. The sequence {xn}naml