MATH 242: Analysis 1 (McGill Final)

Currently Playing: This is the Epsilon Delta Definition of Continuity | 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 The epsilon delta definition of continuity is the end of our quest for a rigorous definition of continuity. All quirks of continuity we have seen are consistent with this definition which mostly comes from the definition of a functional limit. A function f is continuous at a point c if for all epsilon greater than 0, there exists delta greater than 0 so |x-c| less than delta implies |f(x)-f(c)| less than epsilon. In this real analysis lecture we introduce this definition, equivalent definitions, properties of continuity, and a basic epsilon delta proof. #realanalysis Definition of a Functional Limit: https://youtu.be/kVQNhAIFZYc Proof sqrt(x) is Continuous: https://youtu.be/lPWKnynFkcg ★DONATE★ ◆ Support Wrath of Math on Patreon for early access to new videos and other exclusive benefits: https://www.patreon.com/join/wrathofmathlessons ◆ Donate on PayPal: https://www.paypal.me/wrathofmath 0:00 Intro 1:01 Definition 3:25 Why |x-c| isn't Required to be Positive 4:12 When c is not a Limit Point 5:37 Equivalent Definitions of Continuity 7:07 Sequential Characterization of Continuity 8:19 Proving f(x)=x is Continuous using Epsilon Delta Definition of Continuity 9:56 Basic Continuity Laws 11:13 Practice Exercise: Prove sqrt(x) is Continuous