Plasuible Argument that $A$ must be an interval where $A \in R$ be path-connected.

Question

" Let $A\in \Bbb R$ be path-connected.
Give plausible argument that $A$ must be an interval(closed/open/or half-open). Are things as simple in $R^2?$ "

My textbook on analysis mentioned such as above. What is the "plausible argument" that the question intended to ask?

Answer 1

Pick two points in $A$ and consider the set of all points on the path connecting them. What must that set be?