Taylor proves the following (with one incomprehensible step), and Rosenlicht gives it as a problem, a problem!! Is the answer given? Do bears poop in the woods?
Any connected open subset of E^n is arcwise connected.
Definitions from Rosenlicht:
"A metric space E is connected if the only subsets of E which are both open and closed are E and 0. A subset S of a metric space is a connected subset if the subspace S is connected."
"A metric space E is said to be arcwise connected if, given any p,q eE, there is a continuous function f:[0,1] -> E such that f(0)=p , f(1)=q."
Rudin doesn't touch this.