3. If E is a connected subset of M , and if A and B are disjoint open sets in M with E C A U B,provethateitherE C AorE C B. D 6. More generally. if C is a collection of connected subsets of M , all having a point in common, prove that UC is connected. Use this to give another proof that R is connected. Throughout. M denotes an arbitrary metric space with metric d. D 9. [f 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. 1]. An alternate denition of connectedness for metric spaces can be phrased in terms of continuous real-valued functions: Prove that M is disconnected if and only if there is a continuous function f : M > R such that f "([01) = C?) while f"((oo, 0)) -, Q) and f"((0. 00)) a Q. [Hint If A and B are a disconnection, consider f(x) = d(x, A) d(x, 3).] 25. (8) Give an example of a continuous function having a connected range but a dis- connected domain. (b) Let D C IR, and let f : D > R be continuous. Prove that D is connected if {(x. f(.r)) : x e D]. the graph of f. is a connected subset of R2