1. Let (X 1 , d 1 ) and (X 2 , d 2 ) be metric spaces and suppose f : X 1 → X 2 . Mark each statement True or False. Justify each answer. (a) If f is continuous and C is open in X 1 , then f (C ) is...

1. Let (X₁, d₁) and (X₂, d₂) be metric spaces and suppose f : X₁
→ X₂. Mark each statement True or False. Justify each answer.

(a) If f is continuous and C is open in X₁, then f (C ) is open in X₂.

(b) If f is continuous and C is compact in X₁, then f (C ) is compact in X₂.

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

2. Let (X₁, d₁
) and (X₂, d₂) be metric spaces and suppose f : X₁
→ X₂
and g : X₁
→ X₂
are both continuous on X₁. If D ⊆ X1 and f (x) = g (x) for all x ∈ D, prove that f (x) = g (x) for all x ∈ cl D.