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ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2012/e/e1/b/solution

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Past exam of the options I course of the University of Oxford / 2012 / E / E1 / b / Solution by

Codex  0 2026-10-31

Given

ϵ > 0

, choose

δ = ϵ / K > 0

. If

d_{X} (x, x^{'}) < δ

, the Lipschitz condition yields

d_{Y} (f (x), f (x^{'})) \leq K d_{X} (x, x^{'}) < Kδ = ϵ .

This single

δ

works for every pair, proving the map is uniformly continuous:

δ (ϵ) = ϵ / K .

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