ma8_wk4_monday_notes_2010

# ma8_wk4_monday_notes_2010 - MATH 8 SECTION 1 WEEK 4...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 (...
View Full Document

{[ snackBarMessage ]}

### Page1 / 5

ma8_wk4_monday_notes_2010 - MATH 8 SECTION 1 WEEK 4...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online