Generalized Discontinuity of Real-Valued Functions

Zduńczyk, Rafał

doi:10.2478/tmmp-2013-0014

Abstract

We present a proof of the theorem on countability of the set of points of generalized discontinuity of an ( , ) regular real function f:X→ℝ, where is a local system in X and is a partition of X. We start with a definition of a local system in a generalized form and with basic properties of local systems. The concepts are illustrated with examples. The main result is applied both for regularities in the sense of density connected with the Lebesgue measure on ℝⁿ (Lebesgue density) and with Baire category -density), respectively.

References

[1] CIESIELSKI, K. LARSON, L. OSTASZEWSKI, K.: 1-density continuous functions, Mem. Airier. Math. Soc. 107 (1994), p. 133.
Search in Google Scholar
[2] ENE, V.: Real Functions Current Topics, in: Lecture Notes in Math., Vol. 1603, Springer- Verlag, Berlin, 1995.
Search in Google Scholar
[3] FILIPCZAK, T.: Intersection conditions for some density and 1-density local systems, Real Anal. Exchange 15 (1990), 170 192.10.2307/44151997
Search in Google Scholar
[4] GR ANDE, M.: On the sums of unilaterally approximately continuous and approximately jump functions, Real Anal. Exchange 28 (2003), 62.3 6.30.
Search in Google Scholar
[5] GRÄTZER, G.: General Lattice Theory. Birkhäuser Verlag, Basel-Stuttgart, 1978.10.1007/978-3-0348-7633-9
Search in Google Scholar
[6] HOBSON, E. W.: Tlie Theory of Functions of Real Variable and the Theory of Fourier’s Series. Cambridge University Press, London, 1921.
Search in Google Scholar
[7] MAZURKIEWICZ, S.: O pewnej mnogości, która ma z każdą prostą dwa, i tylko dwa punkty wspólne, C. R. Varsovia 7 (1914), .382 .384.
Search in Google Scholar
[8] ŚWIĄT KO W SKI, T.: O pewnym uogólnieniu pojęcia, pochodnej, Zeszyty Nauk. Politech. Lódz. Mat. 149 (1972), 89 101.
Search in Google Scholar
[9] THOMSON, B.: Derivation bases on the real line 11, Peal Anal. Exchange 8 (198.3), 278 442.10.2307/44153415
Search in Google Scholar
[10] THOMSON, B.: Real Functions, in: Lecture Notes in Math., Vol. 1170, Springer-Verlag, 1980.
Search in Google Scholar
[11] WILCZYŃSKI, W.: A category analogue of the density topology, approximate continuity and, the approximete derivative, Peal Anal. Exchange 10 (1984 85), 241 265.10.2307/44153572
Search in Google Scholar
[12] WILCZYŃSKI, W.: Density topologies, in: Handbook of Measure Theory, Vol. I and II, North-Holland, Amsterdam, 2002, pp. 675 702.10.1016/B978-044450263-6/50016-6
Search in Google Scholar
[13] ZAJICEK, L.: Porosity, J-density topology and, abstract density topologies, Real Anal. Exchange 12 (1987), 313 .326.
Search in Google Scholar
[14] ZDUŃCZYK, P.: Unilateral I-approximate limits of real functions, Real Anal. Exchange 34 (2008), 105 114. 10.14321/realanalexch.34.1.0105
Search in Google Scholar

Generalized Discontinuity of Real-Valued Functions

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