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Official problem 5 · Uniform continuity in metric spaces
- Source locator
- Part M, Section 1, official problem 5, paper PDF page 2
- Review note
- Independently derived; lower bound and Lipschitz constants checked; official-key match · 2026-07-28
Problem summary
Classify two reciprocal distance-sum functions by deciding whether each is uniformly continuous on an arbitrary metric space with distinct fixed points.
Core idea
Use the triangle inequality to keep each denominator uniformly away from zero, then combine the reciprocal estimate with Lipschitz bounds for distance functions.
Derivation
- 1The sum of the distances to the first two fixed points is at least the distance between those points, a positive constant.
- 2Adding a third nonnegative distance preserves that lower bound.
- 3A sum of two distance functions is 2-Lipschitz, while a sum of three is 3-Lipschitz.
- 4On values at least δ, the reciprocal map changes by at most |u − v|/δ², so both composed functions are globally Lipschitz.
Final answer
Both functions are uniformly continuous, so the correct choice is option (c).