  We revisit the causal structures $J^+$ and $K^+$ on a spacetime, and
introduce a new one, called $k^+$. The $k^+$-relation can be used to
characterize causal curves, and for smooth Lorentzian metrics, it yields the
same result as the standard $J^+$-relation. If, on the other hand, the metric
is merely continuous, then the different causal structures become inequivalent.
We compare them by investigating three properties, namely the validity of the
push-up lemma, the openness of chronological futures, and the existence of
limit causal curves. Depending on the definition of causal structure chosen, we
show that at most two of these three properties hold for continuous metrics. In
particular, by using the new relation $k^+$, the push-up lemma holds even when
the metric is continuous, while it generally does not for the standard
$J^+$-relation. Finally, we argue that, in general, no reasonable notion of
causal structure can have all three properties.
