The Cosmological Time Functions and Lightlike Rays

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