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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 ?

  1. A path is a continuous function defined on $I, p: I \to X$
  2. A graph is the subset $\{t, p(t)\} \subset I \times X$
  3. 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).

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    $\begingroup$ Yes, I think "curve" is used with both meanings. When we speak of a "continuous curve" or a "differentiable curve" I understand that the function is under discussion. When we speak of the parameterization of a curve, I understand the point set to be intended. Somewhere or other, I once saw the term "trace" used for the point set, but I don't think that is common. $\endgroup$ – saulspatz Oct 12 '18 at 14:34
  • $\begingroup$ @saulspatz: My experience is different; “parameterization” often signifies the function, not the point set of its image — otherwise you can’t tell different parameterizations of the same point set apart. $\endgroup$ – symplectomorphic Oct 12 '18 at 14:37
  • $\begingroup$ Oh, I think you meant “curve” (not “parameterization”) refers to “point set” in the phrase “parameterization of a curve.” Never mind. $\endgroup$ – symplectomorphic Oct 12 '18 at 14:39
  • $\begingroup$ @symplectomorphic Yes, that's what I meant. $\endgroup$ – saulspatz Oct 12 '18 at 14:44
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    $\begingroup$ In agreement with @saulspatz, I think that "curve" is often used interchangeably to mean the image $p(I)$, the function $p$ itself, and the parameterization. In most cases, the context surrounding the use of the word should give the right meaning, and it would be pedantic (though correct) to try to distinguish the meanings. That being said, if your target audience is a bunch of undergraduates, such pedantry is pedagogically useful; and if the distinction is actually really important for your application and context is insufficient to disambiguate, such pedantry is likely necessary. $\endgroup$ – Xander Henderson Oct 12 '18 at 14:51

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