Let ρ₁ be usual distance and ρ₂ the discrete metric on ℝ. X={n∈ℕ:n≥3},Y={n+1/n:n∈ℕ}. Define f=2 on X and f=3 on Y. P:f:(X∪Y,ρ₁)→(ℝ,ρ₁) is uniformly continuous. Q:the same is true with domain metric ρ₂. Choose truth values.
For n≥3 the points n and n+1/n have distance 1/n→0 but image difference 1, violating uniform continuity at ε=1/2. For the discrete domain, choose δ=1/2: distance below δ forces identical points and zero image difference. **B**.
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