Cauchy functional equation and generalized continuity

Pavel Kostyrko

doi:10.2478/v10127-009-0043-4

.blurhash-client-img { display: none !important; }

Cauchy functional equation and generalized continuity

Tatra Mountains Mathematical Publications

Volume 44 (2009): Issue 1 (December 2009)

By: Pavel Kostyrko

Open Access

|Nov 2012

Abstract

There are many kinds of the generalization of continuity. T. ˇSal´at raised the question: Can everywhere discontinuous solution of Cauchy functional equation ƒ(x + y) = ƒ(x) + ƒ(y) be continuous in some generalized sense? The paper deals with this question

References

[1] BORS´IK, J.-DOBOˇS, J.: On simple continuity points, Real Anal. Exchange 16 (1991), 552-558.10.2307/44153736
Search in Google Scholar
[2] DOBOˇS, J.: Simple continuity and cliquishnees, ˇCas. Pˇest. Mat. 112 (1987), 355-358.
Search in Google Scholar
[3] GENTRY, K. R.-HOYLE, H. B.: Somewhat continuous functions, Czechoslovak. Math. J. 21 (1971), 5-12.10.21136/CMJ.1971.100999
Search in Google Scholar
[4] GRANDE, Z.: Sur les fonctions A-continues, Demonstratio Math. 11 (1978), 519-526.
Search in Google Scholar
[5] JONES, F. B.: Connected and disconnected plane sets and the functional equation f(x + y) = f(x) + f(y), Bull. Amer. Math. Soc. 48 (1942), 115-120.10.1090/S0002-9904-1942-07615-4
Search in Google Scholar
[6] KELLUM, K. R.: Sums and limits of almost continuous functions, Colloq. Math. 31 (1974), 125-128.10.4064/cm-31-1-125-128
Search in Google Scholar
[7] KESTELMAN, H.: On the equation f(x + y) = f(x) + f(y), Fund. Math. 34 (1947), 144-147.10.4064/fm-34-1-144-147
Search in Google Scholar
[8] KOSTYRKO, P.-ˇSAL´AT, T.: On functions with closed graphs, ˇCas. Pˇest. Mat. 89 (1984), 426-432. (In Russian)
Search in Google Scholar
[9] KUCZMA, K.: An Introduction to the Theory of Functional Equations and Inequalities, PWN, Warszawa, 1965.
Search in Google Scholar
[10] LUKEˇS, J. A KOL.: Probl´emy z Matematick´e Anal´yzy, SPN, Praha, 1972. (In Czech)
Search in Google Scholar
[11] MARCUS, S.: Sur le fonctions quasicontinues au sens de S. Kempisty, Colloq. Math. 8 (1961), 47-53.10.4064/cm-8-1-47-53
Search in Google Scholar
[12] NEUBRUNNOV´A, A.-ˇSAL´AT, T.: On almost quasicontinuity, Math. Bohem. 117 (1992), 197-205.10.21136/MB.1992.125901
Search in Google Scholar
[13] NOIRI, T.: On weakly continuous mappings, Proc. Amer.Math. Soc. 46 (1974), 120-124.10.1090/S0002-9939-1974-0348698-5
Search in Google Scholar
[14] PREISS, D.: A note on symmetrically continous mappings, ˇCas. Pˇest. Mat. 96 (1971), 262-264.
Search in Google Scholar
[15] ROSEN, H.: On characterizing extentable conectivity functions by associated sets, Real Anal. Exchange 22 (1996-97), 279-283.10.2307/44152750
Search in Google Scholar
[16] SINGAL, M. K.-SINGAL, A. R.: Almost continuous mappings, Yokohama Math. J. 16 (1968), 63-73.
Search in Google Scholar
[17] SM´ITAL, J.-ˇSAL´AT, T.: Remarks on two generalizations of the notion of continuity, Acta Fac. Rer. Nat. Univ. Com. 36 (1980), 115-116.
Search in Google Scholar
[18] STALLINGS, J.: Fixed point theorem for connectivity maps, Fund. Math. 47 (1959), 249-263.10.4064/fm-47-3-249-263
Search in Google Scholar
[19] ˇSAL´AT, T.: Some generalizations of the notion of continuity and Denjoy property of functions, ˇCas. Pˇest. Mat. 99 (1974), 380-385.
Search in Google Scholar
[20] THOMSON, B. S.: Symmetric Properties of Real Functions, Marcel Dekker, New York, 1994.
Search in Google Scholar