4. (a) (10 marks) Let f be integrable on (a,b). Suppose ceR and g: (a +cb+ → R such that g(x) = f(x-c), 2 € (a + c, b +

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4 A 10 Marks Let F Be Integrable On A B Suppose Cer And G A Cb R Such That G X F X C 2 A C B 1 (24.88 KiB) Viewed 17 times

4. (a) (10 marks) Let f be integrable on (a,b). Suppose ceR and g: (a +cb+ → R such that g(x) = f(x-c), 2 € (a + c, b + c Prove that g is integrable on (a +c,b+and 5 [* $(a) dier = EL f) pb+c g(x) dr 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 1,Y ER Show that h(x) = cr for any reR, where c=h(1). (Hint: Fix any r,y e R and integrate h(t + y)=h(t) +h(y) with respect to t on (0,r). Then use (a).)