SCHWEITZER 1998 Problem 9

Let G be a domain (connected open set) in the $R^2$ plane whose boundary is locally connected. Prove that for every point q of the boundary of G there exists a simple arc $v_q$ in which $q\in v_q$ and $v_q\setminus\{q\}\subset G$ .

other questions: (i) Show that local connectedness cannot be replaced by connectedness. (ii) Show that if we replace the $R^2$ plane with $R^3$ space, the statement does not hold.