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Unformatted text preview: { Lecture 14 As before, let f : R R be the map defined by → 0 f ( x ) = e − 1 /x if x ≤ 0, (3.162) if x > 0. This is a Cinf ( R ) function. Take the interval [ a, b ] ∈ R and define the function f a,b : R → R by f a,b ( x ) = f ( x − a ) f ( b − x ). Note that f a,b > 0 on ( a, b ), and f a,b = 0 on R − ( a, b ). We generalize the definition of f to higher dimensions. Let Q ⊆ R n be a rectangle, where Q = [ a 1 , b 1 ] × ··· × [ a n , b n ]. Define a new map f Q : R n R bye → f Q ( x 1 , . . . , x n ) = f a 1 ,b 1 ( x 1 ) . . . f a n ,b n ( x n ) . (3.163) Note that f Q > 0 on Int Q , and that f Q = 0 on R n − Int Q . 3.9 Support and Compact Support Now for some terminology. Let U be an open set in R n , and let f : U R be a → continuous function. Definition 3.26. The support of f is supp f = x ∈ U : f ( x ) = 0 } . (3.164) For example, supp f Q = Q . Definition 3.27. Let f : U R be a continuous function. The function f is → compactly supported if supp...
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This note was uploaded on 10/02/2010 for the course MAT unknown taught by Professor Unknown during the Fall '04 term at MIT.
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