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), X x € (a

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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 X X A 1 (60.72 KiB) Viewed 27 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), X x € (a + c,b+c] Prove that g is integrable on (a + c,b+c] and . obte f(x) dx = g(x) dx a+c (b) (20 marks) Let h: R+R be integrable on every bounded interval and h(x + y) = h(x) +h(y) for any x, y ER Show that h(x) = cx for any x ER, where c=h(1). (Hint: Fix any x, y ER and integrate h(t+y) = h(t) +h(y) with respect to t on [0, x]. Then use (a).)