1. A subset S of is said to be disconnected if there exist disjoint open sets U and V in such that S ⊆ U ∪ V, S ∩ U ≠ ∅, and S ∩ V ≠ ∅. If S is not disconnected, then it is said to be connected....

1. A subset S of

is said to be disconnected if there exist disjoint open sets U and V in

such that S ⊆ U ∪ V, S ∩ U ≠ ∅, and S ∩ V ≠ ∅. If S is not disconnected, then it is said to be connected. Suppose that S is connected and that f :

→

is continuous. Prove that f (S ) is connected. (Hint: Use Corollary 2.15.)

2. Let f : D →
. Mark each statement True or False. Justify each answer.

(a) f is uniformly continuous on D iff for every ε > 0 there exists a δ > 0 such that | f (x) – f (y) | <><>∈ D.

(b) If D = { x }, then f is uniformly continuous at x.

(c) If f is continuous and D is compact, then f is uniformly continuous on D.