Just to add to Kostya's answer above, it's worth noting a couple of facts:
- Causality Implies the Lorentz Group;
- On the structure of causal spaces and A new topology for curved space–time which incorporates the causal, differential, and conformal structures.
The bottom-line is that the Causal Structure differs from the Metric by a conformal transformation. So, effectively (i.e., modulo a conformal transformation), it doesn't matter whether you supply a Causal Structure or a Metric to your manifold — of course, keep in mind that a point at infinity can cause all sorts of trouble (but usually, in Physics, we'll assume things are "well behaved"... ;-).
To directly answer your question, coupling Luboš's and Kostya's answer: yes, causality and continuity (in the sense given by the metric structure) can be thought of as being the same thing modulo a conformal transformation, i.e., as far as physical events are concerned; having said that, note that acausal regions can be continuously connected, you just can't access those regions physically.
Another way to think of this is to 'split' (foliate) your spacetime manifold in three different regions determined by the light-cone, i.e., use your Causal Structure to foliate your manifold into a spacelike region, a timelike region and a lightlike one. Each one of these can be simply-connected (continuous in the metric sense) and understood in terms of the Causal Structure (modulo a conformal transformation).