Unformatted text preview: Suppose a,b,c,k,m ∈ R and a < c < b . Working directly from the deﬁnition show that the piecewise constant function f : [ a,b ] 7→ R deﬁned 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 deﬁnition 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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 Spring '07
 thieme
 Calculus, Continuous function, Advanced Calculus Test

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