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Unformatted text preview: Suppose a,b,c,k,m R and a < c < b . Working directly from the denition show that the piecewise constant function f : [ a,b ] 7 R dened by f ( x ) = k if a x c and f ( x ) = m if c < x b is integrable. 6. a. (Without proof), give an example of an integrable function f : [ a,b ] 7 R such that f 0 and R b a f = 0, but f 6 0. b. Suppose that a < b are real numbers. Working directly from the denition show that if f : [ a,b ] 7 [0 , ) is continuous (and hence integrable), and f 6 0 then R b a f > 0. Bonus: Sketch a proof that every monotone function f : [ a,b ] 7 R is integrable. Hint: How many jumpdiscontinuities of height larger than  f ( b )f ( a )  / N for N Z + can f have?...
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This note was uploaded on 04/02/2009 for the course MAT 371 taught by Professor Thieme during the Spring '07 term at ASU.
 Spring '07
 thieme
 Calculus

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