Qing and Rafi (Morse boundaries I: CAT(0) spaces, 2019. arXiv:1909.02096) introduced the sublinearly contracting boundary for spaces. Every point of this is uniquely represented by a geodesic ray: ray b where every disjoint ball projects to subset whose diameter bounded sublinear function in terms ball’s distance origin. This paper analyzes bahaviour horofunctions associated such rays, example,...