Let X X be a continuously homogeneous Hausdorff continuum. We prove that if there is a sequence A 1 , A 2 , … {A_1},{A_2}, \ldots of its arc components with X = c1 A 1 ∪ c1 A 2 ∪ ⋯ X = {\text {c1}}{A_1} \cup {\text {c1}}{A_2} \cup \cdots , and there is an arc component of X X with nonempty interior, then X X is arcwise connected. As consequences and applications we get: (1) if X X is the countable union of arcwise connected continua, then X X is arcwise connected; (2) if X X is nondegenerate and metric, the number of its arc components is countable and it contains no simple triod, then it is either an arc or a simple closed curve; and, in particular, (3) an arc is the only nondegenerate continuously homogeneous arc-like metric continuum with countably many arc components.