ma8_wk4_monday_notes_2010

ma8_wk4_monday_notes_2010 - MATH 8 SECTION 1 WEEK 4...

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Unformatted text preview: MATH 8, SECTION 1, WEEK 4 - RECITATION NOTES TA: PADRAIC BARTLETT Abstract. These are the notes from Monday, Oct. 18th’s lecture, where we started to discuss the ideas of limits and continuity. 1. Random Question Question 1.1. So, in R 2 , you can draw at most 6 equilateral triangles around a given point; this is a simple consequence of the internal angle of a equilateral triangle being 60 ◦ . A natural generalization of the above question, then, is the following: in R 3 , what is the maximum number of regular tetrahedra can you fit around a given point? 2. Continuity: Definitions Definition 2.1. If f : X → Y is a function between two subsets X,Y of R , we say that lim x → a f ( x ) = L if and only if (1) (vague:) as x approaches a , f ( x ) approaches L . (2) (precise; wordy:) for any distance > 0, there is some neighborhood δ > of a such that whenever x ∈ X is within δ of a , f ( x ) is within of L . (3) (precise; symbols:) ∀ > , ∃ δ > 0 s.t. ∀ x ∈ X, ( | x- a | < δ ) ⇒ ( | f ( x )- L | < ) . Definition 2.2. A function f : X → Y is said to be continuous at some point a ∈ X iff lim x → a f ( x ) = f ( a ) . Somewhat strange definitions, right? At least, the two “rigorous” definitions are somewhat strange: how do these epsilons and deltas connect with the rather simple concept of “as x approaches a , f ( x ) approaches f ( a )”? To see this a bit better, consider the following image: 1 2 TA: PADRAIC BARTLETT a f(a) a+δ f(a)+ϵ f(a)-ϵ a-δ This graph shows pictorially what’s going on in our “rigorous” definition of limits and continuity: essentially, to rigorously say that “as x approaches a , f ( x ) approaches f ( a )”, we are saying that • for any distance around f ( a ) that we’d like to keep our function, • there is a neighborhood ( a- δ,a + δ ) around a such that • if f takes only values within this neighborhood (...
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ma8_wk4_monday_notes_2010 - MATH 8 SECTION 1 WEEK 4...

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