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What are advantages and disadvantages of considering arbitrary functions from a subset of $\mathbb{R}$ to $\mathbb{R}^n$ rather than classically defined paths (that is functions from a connected subset of $\mathbb{R}$ to $\mathbb{R}^n$)?

In other words, why many mathematicians have chosen when they define paths to consider only functions from a connected subset of $\mathbb{R}$?

What results do not generalize nicely to unconnected subsets of $\mathbb{R}$?

One such result is that a function with zero derivative is constant. But we can replace "constant" with "locally constant" and pass.

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    $\begingroup$ I'd not say that the Cantor's set is a "path". $\endgroup$ – ajotatxe Aug 5 '16 at 20:37
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    $\begingroup$ Doesn't a path imply you can get from any point to any other? If you define a path as a map from an arbitrary subset of $\Bbb R$ then I think it would be misnamed as a path. $\endgroup$ – Gregory Grant Aug 5 '16 at 20:47
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    $\begingroup$ You'll be throwing at least the following babies out with the bath water: Cauchy's integral theorem, Riemann surfaces, covering spaces, homotopy theory, ... I.e., essentially everything that depends on "path" having its intuitive meaning as a continuous route from one point in a space to another. $\endgroup$ – Rob Arthan Aug 5 '16 at 20:49
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The first thought that comes to mind is that path-connectedness would no longer imply connectedness, which seems contrary to intuition.

If the subsets of $\mathbb{R}$ are truly allowed to be arbitrary, then every topological space is path connected trivially, since you can take the points $\{a, b\}, a \neq b \in \mathbb{R}$ and for $x,y \in X$ set $f(a)=x$ and $f(b)=y$ and this function is then a "continuous path from x to y."

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