  We consider the usual causal structure $(I^+,J^+)$ on a spacetime, and a
number of alternatives based on Minguzzi's $D^+$ and Sorkin and Woolgar's
$K^+$, in the case where the spacetime metric is continuous, but not
necessarily smooth. We compare the different causal structures based on three
key properties, namely the validity of the push-up lemma, the openness of
chronological futures, and the existence of limit causal curves. Recall that if
the spacetime metric is smooth, $(I^+,J^+)$ satisfies all three properties, but
that in the continuous case, the push-up lemma fails. Among the proposed
alternative causal structures, there is one that satisfies push-up and open
futures, and one that has open futures and limit curves. Furthermore, we show
that spacetimes with continuous metrics do not, in general, admit a causal
structure satisfying all three properties at once.
