 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ib/paper-3/2e/b/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

x \mapsto cos (1/ x)

is uniformly continuous on

[1, \infty)

. Its derivative satisfies

\frac{s i n ( 1/ x )}{x ^{2}} \leq 1,

∣ cos (1/ x) - cos (1/ y) ∣ \leq ∣ x - y ∣.

Thus the function is Lipschitz, which implies uniform continuity.

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