Generalized Derivative and Generalized Continuity

Kupka, Ivan

doi:10.2478/tmmp-2019-0021

Abstract

The main objective of this article is to show that generalized differentiation can be understood as a process of comparing functions and their generalized continuity properties. We show it by working with generalized notions of derivative and continuity. The article covers wide range of types of generalized continuity.

References

[1] AMBROSIO, L.—KIRCHHEIM, B.: Rectifiable sets in metric and Banach spaces,Math. Ann. 318 (2000), 527–555.10.1007/s002080000122
Search in Google Scholar Back to article
[2] AVERSA, V.—PREISS, D.: Lusin’s theorem for derivatives with respect to a continuous function, Proc. Amer. Math. Soc. 127 (1999), 3229–3235.10.1090/S0002-9939-99-05086-8
Search in Google Scholar Back to article
[3] BILET, V.—DOVGOSHEY, O.—KÜÇÜKASLAN, M.: Uniform boundedness of pretan-gent spaces, local constancy of metric valued derivatives and strong right upper porosity at a point,J. Anal. 21 (2013), 31–55.
Search in Google Scholar Back to article
[4] BORSÍK_, J.: Bilateral quasicontinuity in topological spaces, Tatra Mt. Math. Publ. 28 (2004), 159–168.
Search in Google Scholar Back to article
[5] _______ On the points of bilateral quasicontinuity of functions, Real Anal. Exchange 19 (1993/1994), 529–536.10.2307/44152402
Search in Google Scholar Back to article
[6] BORŚIK, J.—HOLÁ, L’.—HOLÝ, D.: Baire spaces and quasicontinuous mappings, Filomat 25 (2011), 69–83.10.2298/FIL1103069B
Search in Google Scholar Back to article
[7] DEIMLING, K.: Nonlinear Functional Analysis. Springer-Verlag, Berlin, 1985.10.1007/978-3-662-00547-7
Search in Google Scholar Back to article
[8] DOVGOSHEY, O.—MARTIO, O.: Tangent spaces to general metric spaces, Rev. Roumaine Math. Pures Appl. 56 (2011), 137–155.
Search in Google Scholar Back to article
[9] ENGELKING, R.: General Topology. PWN, Warszaw, 1977.
Search in Google Scholar Back to article
[10] GROSSMAN, M.—KATZ, R.: Non-Newtonian Calculus (7th ed.), Mathco, Rockport, Massachusetts, 1979.
Search in Google Scholar Back to article
[11] KUPKA, I.: Topological generalization of Cauchy’s mean value theorem, Ann. Acad. Sci. Fenn. Math. Diss. 41 (2016), 315–320.10.5186/aasfm.2016.4120
Search in Google Scholar Back to article
[12] _______ On similarity of functions,Top.Proc. 36 (2010), 173–187.
Search in Google Scholar Back to article
[13] _______ Similar functions and their properties, Tatra Mt. Math. Publ. 55 (2013), 47–56.10.2478/tmmp-2013-0018
Search in Google Scholar Back to article
[14] _______ Measurability of similar functions, Ann. Acad. Sci. Fenn. Math. Diss. 42 (2017), 803–808.10.5186/aasfm.2017.4246
Search in Google Scholar Back to article
[15] LEE, P. Y.—TANG, W. K.—ZHAO, D.: An equivalent definition of functions of the first Baire class, Proc. Amer. Math. Soc. 129 (2013), 47–56.
Search in Google Scholar Back to article
[16] LEVINE, N.: Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36–41.10.1080/00029890.1963.11990039
Search in Google Scholar Back to article
[17] MARCINIAK, M.: Finitely continuous, Darboux functions, Real Anal. Exchange 19 (1993/1994), 394–413.
Search in Google Scholar Back to article
[18] MATEJDES, M.: Quasicontinuous selections of upper Baire continuous mappings,Mat. Vesnik 62 (2010), 69–76.
Search in Google Scholar Back to article
[19] LIGHT, G. L.: An introductory note on relative derivative and proportionality,Int. J. Contemp. Math. Sci. 1 (2006), 327–332.10.12988/ijcms.2006.06034
Search in Google Scholar Back to article
[20] LYTCHAK, A.: Differentiation in metric spaces, Algebra i Analiz 16 (2004), 128–161.10.1090/S1061-0022-05-00888-5
Search in Google Scholar Back to article
[21] O’MALLEY, R. J.: Baire* 1, Darboux functions, Proc. Amer. Math. Soc. 60 (1976), 47–56.10.2307/2041139
Search in Google Scholar Back to article
[22] NEUBRUNN, T.: Quasicontinuity, Real Anal. Exchange 14 (1988–89), 259–306.10.2307/44151947
Search in Google Scholar Back to article
[23] NJÅSTAD, O.: On some classes of nearly open sets, Pacific J. Math. 15 (1965), 961–970.10.2140/pjm.1965.15.961
Search in Google Scholar Back to article
[24] PAWLAK, R. J.: On some class of functions intermediate between the class B*₁and the family of continuous functions, Tatra Mt. Math. Publ. 19 (2010), 135–144.
Search in Google Scholar Back to article
[25] PIOTROWSKI, Z.: A survey of results concerning generalized continuity on topological spaces, Acta Math. Univ. Comenian. 52–53 (1987), 91–110; Algebra i Analiz 16 (2004), 128–161.
Search in Google Scholar Back to article
[26] ROBERTSON, A. P.—ROBERTSON, W.: Topological Vector Spaces. Cambridge University Press, Cambridge, 1964.
Search in Google Scholar Back to article

Generalized Derivative and Generalized Continuity

Abstract

Paradigm

My account