Analysis MOC

Uniform continuity

Uniform continuity¹ is a stronger notion than continuity in metric spaces. A function 𝑓 :𝑋 →𝑌 between metric spaces is uniformly continuous iff for every 𝜖 >0 there exists 𝛿(𝜖) >0 such that anal

𝑑𝑋(𝑥,𝑦)<𝛿(𝜖)⟹𝑑𝑌(𝑓(𝑥),𝑓(𝑦))<𝜖

Properties

• Heine-Cantor theorem — Continuous functions with compact domain are uniformly continuous.

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Footnotes

1. German gleichmäßige Stetigkeit. ↩