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 |xc| 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:    • EpsilonDelta Definition of Functiona...  
Proof sqrt(x) is Continuous:    • Proof: sqrt(x) is Continuous using Ep...  
Sequence Definition of Continuity: (coming soon)
Proving the Basic Continuity Laws: (coming soon)

1:01 Definition
3:25 Why |xc| 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

Proof f(x)=x is Continuous using EpsilonDelta Definition:    • Proof: f(x) = x is Continuous using E...  
Proof |x| is Continuous using EpsilonDelta Definition:    • Proof: f(x) = |x| is Continuous using...  
Proof e^x is Continuous using EpsilonDelta Definition:    • Proof: e^x is Continuous using Epsilo...  
Proof sin(x) is Continuous using EpsilonDelta Definition:    • Proof f(x)=sin(x) is Continuous using...  

Real Analysis Course:    • Real Analysis  
Real Analysis exercises:    • Real Analysis Exercises  

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