Engineering Calculus Notes 332

Engineering Calculus Notes 332 - in S as well as points not...

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320 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION 2. f ( x ) achieves its maximum (resp. minimum) on B ε ( −→ x 0 ) at −→ x = −→ x 0 . A local extremum of f ( x ) is a local maximum or local minimum. To handle sets more complicated than intervals, we need to formulate the analogues of interior points and endponts. Defnition 3.6.8. Let S R 3 be any set in R 3 . 1. A point −→ x R 3 is an interior point of S if S contains some ball about −→ x : B ε ( −→ x ) S. The set of all interior points of S is called the interior of S , denoted int S . A set S is open if every point is an interior point: S = int S . 2. A point −→ x R 3 is a boundary point of S if every ball B ε ( −→ x ) , ε > 0 contains points
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Unformatted text preview: in S as well as points not in S : B ε ( −→ x ) ∩ S n = ∅ , but B ε ( −→ x ) n⊂ S. The set of boundary points of S is called the boundary and denoted ∂S . The following are relatively easy observations (Exercise 10 ): Remark 3.6.9. 1. For any set S ⊂ R 3 , S ⊆ int S ∪ ∂S. 2. The boundary ∂S of any set is closed. 3. S is closed precisely if it contains its boundary points: S closed ⇔ ∂S ⊂ S. 4. S ⊂ R 3 is closed precisely if its complement R 3 \ S := { x ∈ R 3 | x / ∈ S } is open ....
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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