m3114 - Dirichlet Function Functional Limits Continuous...

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Unformatted text preview: Dirichlet Function Functional Limits Continuous Functions Workshop #6 Compact Sets Uniform Continuity The Inter Math 311: Advanced Calculus Wolmer V. Vasconcelos Set 4 Spring 2008 Dirichlet Function Functional Limits Continuous Functions Workshop #6 Compact Sets Uniform Continuity The Inter Main Goal Understand useful functions f : A ⊂ R → R Dirichlet Function Functional Limits Continuous Functions Workshop #6 Compact Sets Uniform Continuity The Inter Building Blocks After building the real number set R , we treated various notions that will be used intensively: 1 Sequences ( a n ) and their limits (or lack of) ( a n ) → a 2 Distinguished subsets of R : neighborhoods, open sets etc 3 If f : R → R and ( a n ) is a sequence then ( f ( a n )) is also a sequence: a n → f → f ( a n ) If ( a n ) is an interesting sequence (what does this mean?) , for what types of functions f will ( f ( a n )) be interesting? Dirichlet Function Functional Limits Continuous Functions Workshop #6 Compact Sets Uniform Continuity The Inter Outline 1 Dirichlet Function 2 Functional Limits 3 Continuous Functions 4 Workshop #6 5 Compact Sets 6 Uniform Continuity 7 The Intermediate Value Theorem 8 The Derivative 9 Mean Value Theorem 10 HomeWork to be Handed in 11 Workshop #7 Dirichlet Function Functional Limits Continuous Functions Workshop #6 Compact Sets Uniform Continuity The Inter Dirichlet Function It might be a good idea to have wonderful functions at hand: 1 (Dirichlet Function) f ( x ) = x ∈ Q 1 x / ∈ Q 2 f ( x ) = x sin ( 1 / x ) x 6 = x = 3 Let f ( x ) be your favorite function: polynomials, rational functions, trig functions, ζ ( x ) ? Dirichlet Function Functional Limits Continuous Functions Workshop #6 Compact Sets Uniform Continuity The Inter Outline 1 Dirichlet Function 2 Functional Limits 3 Continuous Functions 4 Workshop #6 5 Compact Sets 6 Uniform Continuity 7 The Intermediate Value Theorem 8 The Derivative 9 Mean Value Theorem 10 HomeWork to be Handed in 11 Workshop #7 Dirichlet Function Functional Limits Continuous Functions Workshop #6 Compact Sets Uniform Continuity The Inter Functional Limit Now we look at a notion at the root of Calc: Definition Let f : A → R , and let c be a limit point of the domain A . We say lim x → c f ( x ) = L provided that, for all > 0, there exists δ > such that whenever 0 < | x- c | < δ (and x ∈ A ) it follows that | f ( x )- L | < . Let us walk through the functional limit template: 1 Let f : A → R , c limit point of A 2 Given > 0 that is, arbitrary 3 There is δ > 0 that is, δ is a function of and c 4 Such that < | x- c | < δ ⇒ | f ( x )- L | < Dirichlet Function Functional Limits Continuous Functions Workshop #6 Compact Sets Uniform Continuity The Inter Example f ( x ) = x sin ( 1 / x ) x 6 = x = Let us examine its continuity at c = 0: Let > | f ( x )- f ( ) | = | x sin ( 1 / x )- | ≤ | x | Thus if we choose δ = , whenever | x- | = | x | < δ , | f ( x )- f ( ) | < ....
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This note was uploaded on 04/02/2008 for the course MATH 311 taught by Professor Vasconceles during the Spring '08 term at Rutgers.

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m3114 - Dirichlet Function Functional Limits Continuous...

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