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

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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 ) > 0 for some x 0 [ 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. b. Let f, g ∈ R [ a, b ] be such that f ( x ) = g ( x ) for all but a finite number of points in [ a, b ]. Prove that R b a f = R b a g . c. Is the result in part (a) still true if f ( x ) = 0 for all but countably many points in [ a, b ]?

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