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: https://www.youtube.com/playlist?list=PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli Real Analysis exercises: https://www.youtube.com/playlist?list=PLztBpqftvzxXAN05Gm3iNmpz9SkVfLNqC Uniform Continuity on Compact Sets Proof: https://youtu.be/YfppcboF7aA Sequential Criterion for Lack of Uniform Continuity Proof: https://youtu.be/L9XZc9m4yXg Get the textbook for this course! https://amzn.to/3CMdgjI ★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 0:27 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