1. Let X1, X2, and X3 be metric spaces and suppose f : X1 → X2 is uniformly continuous on X1 and g : X2 → X3 is uniformly continuous on X2. Prove that g o f is uniformly continuous on X1.2. Let X1, X2, and X3 be metric spaces with X2 compact. Suppose f : X1 → X2 and g : X2 → X3, with g being bijective and continuous on X2. Define h = g o f .(a) Prove that f is continuous if h is continuous.(b) Prove that f is uniformly continuous if h is uniformly continuous

1. Let X 1 , X 2 , and X 3 be metric spaces and suppose f : X 1 → X 2 is uniformly continuous on X 1 and g : X 2 → X 3 is uniformly continuous on X 2 . Prove that g o f is uniformly continuous on X 1...

1. Let X₁, X₂, and X₃
be metric spaces and suppose f : X₁
→ X₂
is uniformly continuous on X₁
and g : X₂
→ X₃
is uniformly continuous on X₂. Prove that g o f is uniformly continuous on X₁.

2. Let X₁, X₂, and X₃
be metric spaces with X₂
compact. Suppose f : X₁
→ X₂
and g : X₂
→ X₃, with g being bijective and continuous on X₂. Define h = g o f .

(a) Prove that f is continuous if h is continuous.

(b) Prove that f is uniformly continuous if h is uniformly continuous

May 05, 2022

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1. Let X 1 , X 2 , and X 3 be metric spaces and suppose f : X 1 → X 2 is uniformly continuous on X 1 and g : X 2 → X 3 is uniformly continuous on X 2 . Prove that g o f is uniformly continuous on X 1...

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