The definition of a path as a continuous map $I \rightarrow X$ is a completely natural one. But this raises two questions in my mind. First, what properties of the interval give rise to useful topological invariants? Things like compact, metric, complete and ordered all seem fairly important but it's easy to ignore where they're being relied upon (precisely because the definition of path is so intuitive).
The obvious follow up question would be, can we replace $I$ with some other space and still develop an interesting theory? I guess you could go either way, either weakening or strengthening the definition of path.