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Uniform Continuity over Compact Sets

Here we talk about continuity and its connections with compactness. Compactness was first defined around the same time as metric spaces, by Maurice Frechet, and they are more or less a generalization of finite sets. Their interactions with continuous functions have opened up the entire field of functional analysis over the past century, so let's look at the interactions between these important objects. To try everything Brilliant has to offer—free—for a full 30 days, visit https://brilliant.org/ThatMathThing/ . The first 200 of you will get 20% off Brilliant’s annual premium subscription. (This video was sponsored by Brilliant) //Books Rudin - https://amzn.to/3KeL0IC Dummit and Foote Abstract Algebra https://amzn.to/3QxGvwN Folland Real Analysis https://amzn.to/3dCuCao Hungerford Algebra https://amzn.to/3pIJyGV Pedersen Analysis NOW https://amzn.to/3c27MZo //Watch Next Real Analysis Survival Guide    • The Real Analysis Survival Guide   This is why you have imposter syndrome    • This is WHY you have Imposter Syndrome   SoME2 Entry    • The Mathematics of Radio Transmissions   Real Analysis: How to think of Real Numbers    • Intro to Real Analysis: How to think ...   The Banach Fixed Point Theorem    • Fixed Point Iteration - Loki SLAPPED ...   The Real Analysis Survival Guide    • The Real Analysis Survival Guide   Control Theory: The Analyticity of the Laplace transform    • Morera's Theorem in Practice - Laplac...   Introduction to Control Theory    • Introduction to Control Theory   //Music Most music provided by Epidemic Sound with the exception of one song by Tom Fox at Johnny Harris' Patreon Use this referral link to get a 30 day free trial with Epidemic Sound for your YouTube channel: https://www.epidemicsound.com/referra... //Recording Equipment Canon SL3: https://amzn.to/3nZ11KU Canon T6i: https://amzn.to/3FUpkQh Rode VideoMic: https://amzn.to/3lhldGa Blue Yeti Microphone: https://amzn.to/3I1y88N Yeti Nano Microphone: https://amzn.to/3I1mriA SanDisc 256GB SD Card: https://amzn.to/3E3LOOr Neewer 5600K USB LED Lights: https://amzn.to/3xvB9cN Neewer 18 inch Ring Light: https://amzn.to/2ZvgCsc Camera Power Adapter: https://amzn.to/3D3upUu DISCLAIMER: The links above in this description may be affiliate links. If you make a purchase with the links provided I may receive a small commission, but with no additional charge to you :) Thank you for supporting my channel so that I can continue to produce mathematics content for you! 0:00 Start 1:14 Maurice Frechet 1:58 Motivation through Finite Sets 3:36 Brilliant.org Sponsorship 5:37 Continuity via Open Sets 11:01 Continuous Images of Compact Sets 12:51 Uniform Continuity and Heine's Theorem 18:37 Frechet Not Appreciated 19:38 Maximum value on Compact Sets 24:03 Wrap Up