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: • Epsilon-Delta Definiti...
Proof sqrt(x) is Continuous: • Proof: sqrt(x) is Cont...
Sequence Definition of Continuity: (coming soon)
Proving the Basic Continuity Laws: (coming soon)
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
Proof f(x)=x is Continuous using Epsilon-Delta Definition: • Proof: f(x) = x is Con...
Proof |x| is Continuous using Epsilon-Delta Definition: • Proof: f(x) = |x| is C...
Proof e^x is Continuous using Epsilon-Delta Definition: • Proof: e^x is Continuo...
Proof sin(x) is Continuous using Epsilon-Delta Definition: • Proof f(x)=sin(x) is C...
Real Analysis Course: • Real Analysis
Real Analysis exercises: • Real Analysis Exercises
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