MATH 242: Analysis 1 (McGill Final)

Currently Playing: 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! https://www.youtube.com/channel/UCyEKvaxi8mt9FMc62MHcliw/join 🛍 Get the coolest math clothes in the world! https://mathshion.com/ Real Analysis course: https://www.youtube.com/playlist?list=PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli Real Analysis exercises: https://www.youtube.com/playlist?list=PLztBpqftvzxXAN05Gm3iNmpz9SkVfLNqC 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: https://youtu.be/YfppcboF7aA Sequential Criterion for Lack of Uniform Continuity Proof: https://youtu.be/L9XZc9m4yXg ★DONATE★ ◆ Support Wrath of Math on Patreon: https://www.patreon.com/join/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