Often, a continuous function $f: \mathbb{R} \rightarrow M$ from the real numbers $\mathbb{R}$ to a metric space $M$ is called continuous at $r \in \mathbb{R}$ if and only if, for any $\epsilon>0$, there exists some $\delta <0$ such that for all $r' \in [r+\delta, r-\delta]$, the metric distance $d(f(r), f(r'))$ between $f(r)$ and $f(r')$ satisfies $d(f(r), f(r'))< \epsilon$. That is, arbitrarily small changes in the output of a function can be obtained from sufficiently small changes to its input.
But for me, that doesn't feel like the right definition if we are dealing with a Lorentzian manifold. For example, suppose I fire a pulse of light at point $p$ on earth, and you receive it at point $q$ on mars. Since a null curve connects $p$ and $q$, the metric distance between them is 0. Nevertheless, it seems that a curve that hops straight from $p$ to $q$ should not be called continuous, since it "jumps over" so many points in the manifold. Is my intuition wrong? Or should we perhaps be requiring that arbitrarily small changes to the coordinates of the curve in a given reference frame (rather than the metric distance between two points) can be obtained by arbitrarily small changes to the parameter?
Both specific answers and suggestions of good references that deal specifically and carefully with curves in Lorentzian manifolds would be greatly appreciated:)