Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes

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 $M=\mathbb{R}\times \mathcal{S}$, $g= - \beta(\tau,x) d\tau^2 + \bar g_\tau $,(b)ifaspacetime$M$ admits a (continuous) time function $t$ (i.e., it is stably causal) then it admits a smooth (time) function $\tau$ with timelike gradient $\mathrm{\nabla }\tau$ on all $M$.