<abstract xmlns="http://www.w3.org/1999/xhtml"><p> We present a proof of the theorem on countability of the set of points of generalized discontinuity of an (<inline-formula><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="../graphic/tmmp-2013-0014-1.png"></inline-graphic></inline-formula> , <inline-formula><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="../graphic/tmmp-2013-0014-2.png"></inline-graphic></inline-formula>) regular real function f:X→ℝ, where <inline-formula><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="../graphic/tmmp-2013-0014-3.png"></inline-graphic></inline-formula> is a local system in X and <inline-formula><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="../graphic/tmmp-2013-0014-4.png"></inline-graphic></inline-formula> 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 ℝ<sup>n</sup> (Lebesgue density) and with Baire category <inline-formula><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="../graphic/tmmp-2013-0014-5.png"></inline-graphic></inline-formula>-density), respectively. </p></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.

Generalized Discontinuity of Real-Valued Functions

Chair of Real Functions Faculty of Mathematics and Computer Science Łódź University 22 Banacha st. PL–90-238 Łódź POLAND

Tatra Mountains Mathematical Publications

{"article-title":"Generalized Discontinuity of Real-Valued Functions"}

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...

Generalized Discontinuity of Real-Valued Functions

Publié en ligne: 13 août 2013

Pages: 1 - 16

DOI: https://doi.org/10.2478/tmmp-2013-0014

Mots clés
local system, filtering system, generalized limits, density point -density point, lower density

This content is open access.

Generalized Discontinuity of Real-Valued Functions

Publié en ligne: 13 août 2013

Pages: 1 - 16

DOI: https://doi.org/10.2478/tmmp-2013-0014

Mots cléslocal system, filtering system, generalized limits, density point -density point, lower density

This content is open access.

Mots clés
local system, filtering system, generalized limits, density point -density point, lower density