The Cosmological Time Functions and Lightlike Rays

It is proved that all discontinuity points of a finite cosmological time function, $$\tau $$ , are on past lightlike rays. As result, it if (M, g) chronological space-time without rays then there representation g such its function regular. In addition, by reducing conditions regularity sufficient for causal simplicity and pseudoconvexity given. also the second condition can be reduced to satisfies only inextendible past-directed space-time, conformal with an open subspace Minkowski or continuous.