  It is proved that all discontinuity points of a finite cosmological time
function, $\tau$, 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 $\tau$ be continuous.
