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.
