Uniform Continuity Explained | Real Analysis

We introduce the definition of uniform continuity. A function being uniformly continuous on a set is a strictly stronger condition than standard continuity. We'll see some examples illuminating the difference between uniform continuity and standard continuity, do some examples of proving a function is uniformly continuous, and prove a function isn't uniformly continuous. We'll also see two relevant theorems. One states a function continuous on a compact set is uniformly continuous on that set, and the other is a sequential criterion for a function not being uniformly continuous. #realanalysis

Real Analysis Course:    • Real Analysis  
Real Analysis exercises:    • Real Analysis Exercises  

Uniform Continuity on Compact Sets Proof:    • Uniform Continuity on Compact Sets | ...  
Sequential Criterion for Lack of Uniform Continuity Proof:    • Proof: Sequential Criterion for Absen...  

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0:00 Intro
0:27 Enlightening Continuity Proofs
4:36 Definition of Uniform Continuity
7:03 x^2 is Uniformly Continuous on this Compact Set
8:18 Compactness
9:05 Sequential Criterion for Lack of Uniform Continuity