Generalized discountinuity of real-valued funtions

Abstract

We present a proof of the theorem on countability of the set of points of generalized discontinuity of and $(\mathcal{S}$,  $\mathcal{Y}$-regular real function$f\colon X \to \mathbb{R}$, where $(\mathcal{S}$ is a local system in $X$ and$\mathcal{Y}$ 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 $\mathbb{R}^n$ (Lebesgue density) and with Baire category ($\mathcal{I}$-density), respectively.