MATH 242: Analysis 1 (McGill Final)

Currently Playing: Proof: Sequential Criterion for Absence of Uniform Continuity | Real Analysis

Support the production of this course by joining Wrath of Math to access all my Real Analysis videos plus lecture notes at the premium tier! https://www.youtube.com/channel/UCyEKvaxi8mt9FMc62MHcliw/join 🛍 Check out my math fashion brand! 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/4lNpyNN Business Inquiries: wrathofmathlessons@gmail.com We prove a theorem stating a sequential criterion for a function to fail to be uniformly continuous. It is both necessary and sufficient, as we'll prove the equivalence. A function f on a set A fails to be uniformly continuous on A if and only if A contains two sequences xn and yn whose terms get arbitrarily close together, but where f(xn)-f(yn) is always greater than or equal to some fixed positive number. #realanalysis Uniform Continuity Explained: https://youtu.be/MmxtJwr3Kws ★DONATE★ ◆ Support Wrath of Math on Patreon: https://www.patreon.com/join/wrathofmathlessons ◆ Donate on PayPal: https://www.paypal.me/wrathofmath Outro music by Ben Watts: https://crayon-angel.bandcamp.com/music Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● TikTok: https://www.tiktok.com/@wrathofmathedu ● X: https://x.com/wrathofmathedu 0:00 Intro 0:58 Forward Direction 2:41 Reverse Direction 4:02 Conclusion