Uniform Continuity Explained | Real Analysis

Support the production of this course by joining Wrath of Math to access all my real analysis videos plus the lecture notes at the premium tier!    / @wrathofmath   🛍 Get the coolest math clothes in the world! https://mathshion.com/ Real Analysis course:    • Real Analysis   Real Analysis exercises:    • Real Analysis Exercises   Get the textbook! https://amzn.to/45kcMjq 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 Uniform Continuity on Compact Sets Proof:    • Uniform Continuity on Compact Sets | Real ...   Sequential Criterion for Lack of Uniform Continuity Proof:    • Proof: Sequential Criterion for Absence of...   ★DONATE★ ◆ Support Wrath of Math on Patreon:   / wrathofmathlessons   ◆ Donate on PayPal: https://www.paypal.me/wrathofmath 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