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m3114 - Dirichlet Function Functional Limits Continuous...

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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

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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?

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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 ) = 0 x Q 1 x / Q 2 f ( x ) = x sin ( 1 / x ) x 6 = 0 0 x = 0 3 Let f ( x ) be your favorite function: polynomials, rational functions, trig functions, ζ ( x ) ?

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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 > 0 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 0 < | x - c | < δ ⇒ | f ( x ) - L | <

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Dirichlet Function Functional Limits Continuous Functions Workshop #6 Compact Sets Uniform Continuity The Inter Example f ( x ) = x sin ( 1 / x ) x 6 = 0 0 x = 0 Let us examine its continuity at c = 0: Let > 0 | f ( x ) - f ( 0 ) | = | x sin ( 1 / x ) - 0 | ≤ | x | Thus if we choose δ = , whenever | x - 0 | = | x | < δ , | f ( x ) - f ( 0 ) | < . Thus f ( x ) is continuous at x = 0.
Dirichlet Function Functional Limits Continuous Functions Workshop #6 Compact Sets Uniform Continuity The Inter Example Let f ( x ) = 2 x + 1.

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