# zbMATH — the first resource for mathematics

Sequential definitions of continuity for real functions. (English) Zbl 1040.26001
A function $$f:\mathbb{R}\to \mathbb{R}$$ is continuous at a point $$p\in\mathbb{R}$$ if and only if $$\lim_{n\to\infty} x_n= p$$ implies that $$\lim_{n\to\infty}f(x_n)= f(p)$$. If we replace here the usual convergence by another type of convergence we get “a new type of continuity”. In 1948, H. Robbins and E. C. Buck replaced the usual convergence by the Cesàro limitation and obtained so the “Cesàro continuity” which gives the linearity of $$f$$ [Am. Math. Mon. 53, 470–471 (1946); 55, 36 (1948), Problem 4216]. In the following years several authors continued in this study and obtained new types of continuity that led in most cases to the linearity or usual continuity of $$f$$. The authors of this paper unify these types of continuity into the concept of $$G$$-continuity, where $$G$$ is a linear functional. For $$G$$ they take $$A$$-limitation ($$A$$ being a matrix), almost convergence and statistical convergence. The relations of $$G$$-continuities to linearity and usual continuity are investigated in this paper.

##### MSC:
 26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
Full Text:
##### References:
 [1] J. Antoni, On the $$A$$-continuity of real function II, Math. Slovaca 36 (1986), 283-288. · Zbl 0615.40002 [2] J. Antoni and T. Šalát, On the $$A$$-continuity of real functions , Acta Math. Univ. Comenian. 39 (1980), 159-164. · Zbl 0519.40006 [3] S. Banach, Théorie des opérations linéaires , Mono. Mat. I, PWN, Warsaw, 1932. · Zbl 0005.20901 [4] G. Bennett and N.J. Kalton, Consistency theorems for almost convergence , Trans. Amer. Math. Soc. 198 (1974), 23-43. · Zbl 0301.46005 [5] J. Boos, Classical and modern methods in summability , Oxford University Press, Oxford, 2000. · Zbl 0954.40001 [6] J. Borsík and T. Šalát, On $$F$$ · Zbl 0788.26004 [7] R.C. Buck, Solution of problem 4216, Amer. Math. Monthly 55 (1948), 36. [8] ——–, An addendum to “A note on subsequences” , Proc. Amer. Math. Soc. 7 (1956), 1074-1075. JSTOR: · Zbl 0077.27502 [9] J. Connor, The statistical and strong $$p$$-Cesàro convergence of sequences , Analysis 8 (1988), 47-63. · Zbl 0653.40001 [10] ——–, Two valued measures and summability , Analysis 10 (1990), 373-385. · Zbl 0726.40009 [11] ——–, A topological and functional analytic approach to statistical convergence , in Analysis of divergence (W.O. Bray and C.V. Stanojević, eds.), Birkhäuser, Boston, 1999. · Zbl 0915.40002 [12] K. Demirci, A criterion for $$A$$-statistical convergence , Indian J. Pure Appl. Math. 29 (1998), 559-564. · Zbl 0914.40001 [13] H. Fast, Sur la convergence statistique , Colloq. Math. 2 (1951), 241-244. · Zbl 0044.33605 [14] J.A. Fridy, On statistical convergence , Analysis 5 (1985), 301-313. · Zbl 0588.40001 [15] ——–, Statistical limit points , Proc. Amer. Math. Soc. 118 (1993), 1187-1192. · Zbl 0776.40001 [16] G.H. Hardy, Divergent series , Clarendon Press, Oxford, 1949. · Zbl 0032.05801 [17] T.B. Iwiński, Some remarks on Toeplitz methods and continuity , Comment. Math. Prace Mat. 16 (1972), 37-43. · Zbl 0243.40005 [18] G.G. Lorentz, A contribution to the theory of divergent sequences , Acta Math. 80 (1948), 167-190. · Zbl 0031.29501 [19] G.G. Lorentz and K. Zeller, Strong and ordinary summability , Tôhoku Math. J. 15 (1963), 315-321. · Zbl 0185.13301 [20] I.J. Maddox, Elements of functional analysis , Cambridge University Press, Cambridge, 1970. · Zbl 0193.08601 [21] E. Öztürk, On almost-continuity and almost-$$A$$ continuity of real functions , Comm. Fac. Sci. Univ. Ankara Sér. A$$_1$$ Math. 32 (1983), 25-30. · Zbl 0597.40001 [22] G.M. Petersen, Regular matrix transformations , McGraw-Hill, London, 1966. · Zbl 0159.35401 [23] E.C. Posner, Summability-preserving functions , Proc. Amer. Math. Soc. 12 (1961), 73-76. JSTOR: · Zbl 0097.04602 [24] H. Robbins, Problem 4216, Amer. Math. Monthly 53 (1946), 470-471. [25] E. Savas, On invariant-continuity and invariant-$$A$$ continuity of real functions , J. Orissa Math. Soc. 3 (1984), 83-88. · Zbl 0628.26002 [26] E. Savaş and G. Das, On the $$\mathcal A$$ · Zbl 0880.26006 [27] I.J. Schoenberg, The integrability of certain functions and related summability methods , Amer. Math. Monthly 66 (1959), 361-375. JSTOR: · Zbl 0089.04002 [28] E. Spigel and N. Krupnik, On the $$A$$-continuity of real functions , J. Anal. 2 (1994), 145-155. · Zbl 0809.26002 [29] V.K. Srinivasan, An equivalent condition for the continuity of a function , Texas J. Sci. 32 (1980), 176-177. [30] K. Zeller, Über die Darstellbarkeit von Limitierungsverfahren mittels Matrixtransformationen , Math. Z. 59 (1953), 271-277. · Zbl 0052.05502 [31] K. Zeller and W. Beekmann, Theorie der Limitierungsverfahren , 2nd ed., Springer, Berlin 1970. · Zbl 0199.11301
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.