5. If E and F are connected subsets of M with E n F * O, show that E U F is connected. 8. If E and F are nonempty subsets of M, and if E U F is connected, show that EnFO. Throughout, M denotes an arbitrary metric space with metric d. 9. If A C B C A C M, and if A is connected, show that B is connected. In particular, A is connected. 10. True or false? If A C B C C C M, where A and C are connected, then B is connected. 12. If M is connected and has at least two points, show that M is uncountable. [Hint: Find a nonconstant, continuous, real-valued function on M.] 26. Let f : [0, 1 ] - R be defined by f (x) = sin(1/x) for x # 0 and f(0) = 0. Show that although f is not continuous, the graph of f is a connected subset of R?. [ Hint: Use Exercise 9.]