# Local approach to order continuity in Ces\`aro function spaces

## TL;DR

This paper characterizes points of order continuity in Cesàro function spaces built over symmetric spaces, providing conditions for when these spaces inherit order continuity and applying results to specific Cesàro spaces.

## Contribution

It offers a complete characterization of order continuity points in Cesàro function spaces over symmetric spaces, including new criteria and equivalences.

## Key findings

- $(CX)_a = C(X_a)$ under certain conditions
- Order continuity of $X$ implies order continuity of $CX$ when the Cesàro operator is bounded
- Criteria for order continuity points in Cesàro-Orlicz, Lorentz, and Marcinkiewicz spaces

## Abstract

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