4. (a) (10 marks) Let f be integrable on (a, b). Suppose c ER and g: (a + c, b + c + R such that g(x) = f(x-c), rela+c,b

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4 A 10 Marks Let F Be Integrable On A B Suppose C Er And G A C B C R Such That G X F X C Rela C B 1 (39.79 KiB) Viewed 22 times

4. (a) (10 marks) Let f be integrable on (a, b). Suppose c ER and g: (a + c, b + c + R such that g(x) = f(x-c), rela+c,b+c] Prove that g is integrable on (a + c, b + c) and obte Ism f(x) dx = $*** g(x) de late (b) (20 marks) Let h: R+R be integrable on every bounded interval and h(x + y) = h(2) +h(y) for any r, y ER Show that h(1) = cr for any r ER, where c=h(1). (Hint: Fix any 2, y e R and integrate h(t + y) = h(t) +h(y) with respect to t on (0,x]. Then use (a).) 1