  Let $(M,g)$ be a time oriented Lorentzian manifold and $d$ the Lorentzian
distance on $M$. The function $\tau(q):=\sup_{p< q} d(p,q)$ is the cosmological
time function of $M$, where as usual $p< q$ means that $p$ is in the causal
past of $q$. This function is called regular iff $\tau(q) < \infty$ for all $q$
and also $\tau \to 0$ along every past inextendible causal curve. If the
cosmological time function $\tau$ of a space time $(M,g)$ is regular it has
several pleasant consequences: (1) It forces $(M,g)$ to be globally hyperbolic,
(2) every point of $(M,g)$ can be connected to the initial singularity by a
rest curve (i.e., a timelike geodesic ray that maximizes the distance to the
singularity), (3) the function $\tau$ is a time function in the usual sense, in
particular (4) $\tau$ is continuous, in fact locally Lipschitz and the second
derivatives of $\tau$ exist almost everywhere.
