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.
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."