  In general relativity, time functions are crucial objects whose existence and
properties are intimately tied to the causal structure of a spacetime and also
to the initial value formulation of the Einstein equations. In this work we
establish all fundamental classical existence results on time functions in the
setting of Lorentzian (pre-)length spaces (including causally plain continuous
spacetimes, closed cone fields and even more singular spaces). More precisely,
we characterize the existence of time functions by $K$-causality, show that a
modified notion of Geroch's volume functions are time functions if and only if
the space is causally continuous, and lastly, characterize global hyperbolicity
by the existence of Cauchy time functions, and Cauchy sets. Our results thus
inevitably show that no manifold structure is needed in order to obtain
suitable time functions.
