Given a "space" X, at least topological so that continuity has meaning (perhaps a vector space, $R^n$, or the complex plane), and a closed interval I of the real line would it be reasonable to assert the following unambiguous definitions in a mathematical text ?
- A path is a continuous function defined on $I, p: I \to X$
- A graph is the subset $\{t, p(t)\} \subset I \times X$
- A curve is the image $p(I) \subset X$
The first and second definitions seem generally accepted in topology, though the interval is often restricted to $[0, 1]$. But the term curve seems to be used ambiguously with either the meaning above, the above definition for path, or to mean the curve and its path (parameterisation).