HW1 - ANSWERS TO SELECTED EXERCISES: HW1 Exercise 6.1.7 a....

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Unformatted text preview: ANSWERS TO SELECTED EXERCISES: HW1 Exercise 6.1.7 a. Suppose f is continuous on [ a,b ] with f ( x ) > 0 for all x ∈ [ a,b ]. If R b a f = 0, prove that f ( x ) = 0 for all x ∈ [ a,b ] b. Show by example that the conclusion may be false if f is not continuous. Proof : a. We will prove an equivalent statement: if f ( x ) > 0 for some x ∈ [ a,b ], then R b a f > 0. If f ( x ) > 0 for some x ∈ [ a,b ], then by continuity of f there is some positive m > 0 and a nondegenerate subinterval of [ a,b ], say [ c,d ], such that f ( x ) > m for all x ∈ [ c,d ]. But then R b a f > R d c f > m ( d- c ) > 0, and the proof is completed. b. Just consider the function f that satisfies f ( a ) = 1 and f ( x ) = 0 for all x ∈ ( a,b ] (see also next exercise). Exercise 6.2.3 a. Let f be a real-valued function on [ a,b ] such that f ( x ) = 0 for all x 6 = c 1 ,...,c n . Prove that f ∈ R [ a,b ] with R b a f = 0....
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This homework help was uploaded on 04/16/2008 for the course MATH 131B taught by Professor Hitrik during the Winter '08 term at UCLA.

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HW1 - ANSWERS TO SELECTED EXERCISES: HW1 Exercise 6.1.7 a....

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