  The folk questions in Lorentzian Geometry, which concerns the smoothness of
time functions and slicings by Cauchy hypersurfaces, are solved by giving
simple proofs of: (a) any globally hyperbolic spacetime $(M,g)$ admits a smooth
time function $\tau$ whose levels are spacelike Cauchy hyperfurfaces and, thus,
also a smooth global splitting $M= \R \times {\cal S}$, $g= - \beta(\tau,x)
d\tau^2 + \bar g_\tau $, (b) if a spacetime $M$ admits a (continuous) time
function $t$ (i.e., it is stably causal) then it admits a smooth (time)
function $\tau$ with timelike gradient $\nabla \tau$ on all $M$.
