Assume that f: [a, b] → R are integrable. Then prove that for any k ER, kf is also integrable on [a, b], and k.f(x)dx=k. [se a f(x)dx Let P be any partition of [a, b]. (a) For k > 0, show that U(kf, P) = kU (ƒ, P) and L(k · f, P) = kL(ƒ, P). (b) For k < 0, show that U(kf, P) = kL(f, P) and L(k · f, P) = kU (ƒ, P). (c) Since f is integrable on [a, b], by HW5, Exercise 7 we know that there exists a sequence Pn of partitions of [a, b] such that * f(x)dx = lim U (f, P,) = lim L(f, Pm). n-700 12-00 Use this as a fact along with part(a) to prove that Sk. k.f(x)dx=k. k. [° S f(x)dx

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