Local approach to order continuity in Cesà ro function spaces

The goal of this paper is to present a complete characterization of points of order continuity in abstract Cesàro function spaces CX for X being a symmetric function space. Under some additional assumptions mentioned result takes the form (CX)a=C(Xa). We also find simple equivalent condition for this equality which in the case of I=[0,1] comes to X≠L∞. Furthermore, we prove that X is order continuous if and only if CX is, under assumption that the Cesàro operator is bounded on X. This result is applied to particular spaces, namely: Cesàro-Orlicz function spaces, Cesàro-Lorentz function spaces and Cesàro-Marcinkiewicz function spaces to get criteria for OC-points.