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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+b+c P
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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 B C P 1 (18.45 KiB) Viewed 31 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+b+c Prove that g is integrable on [a+c,b+c] and b+c [ f(x) dx - 9(2) de dz= Jate (b) (20 marks) Let h: RR be integrable on every bounded interval and h(x + y)=h(z) +h(y) for any r, yER Show that h(z)= er for any z ER, where c= =h(1). (Hint: Fix any z, y ER and integrate h(t+y)=h(t) +h(y) with respect to t on [(0,r). Then use