08wHW1 - MAT 125B Homework 1 M.Fukuda Please submit your...

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Unformatted text preview: MAT 125B Homework 1 M.Fukuda Please submit your answers at the discussion session on January 22 Tuesday. You can use theorems in the lecuture unless otherwise stated. When you do so write those statements clearly instead of quoting them by numbers. 1. Prove that 1 x E = {1/n : n N} f (x) = 0 x [0, 1] \ E is integrable on [0, 1] by using the definition of integrability (Def 5.9). 2. i) Let f : [a, b] R be bounded. Then, prove, by using the definition of (U) (Def 5.13) that b c b f (x)dx (U) a f (x)dx = (U) a f (x)dx + (U) c f (x)dx for any c (a, b). ii) Suppose f is integrable on [a, b]. Then, prove that f is integrable on [a, c] and [c, b], and b c b f (x)dx = a a f (x)dx + c f (x)dx. You can assume a similar result for (L) f (x)dx. Hint: Use Thm5.15. 3. Let f, g : [a, b] R be integrable on [a, b]. Then, prove that h(x) = max{f (x), g(x)} for x [a, b] is integrable. Hint: Use some theorems in the lecture. 1 ...
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