4. (a) (10 marks) Let f be integrable on [a, b]. Suppose c E R and g : (a + c,b+c] → R such that g(x) = f(x – c), IE (a

[answerhappygod]

4 A 10 Marks Let F Be Integrable On A B Suppose C E R And G A C B C R Such That G X F X C Ie A 1 (87.03 KiB) Viewed 22 times

4. (a) (10 marks) Let f be integrable on [a, b]. Suppose c E R and g : (a + c,b+c] → R such that g(x) = f(x – c), IE (a + c,b+c] X = Prove that g is integrable on (a + c,b+c] and 6 b+c [) = ** g() de f(x) dx atc (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 E R, where c= =h(1). (Hint: Fix any X,Y E R and integrate h(t + y) = h(t) +h(y) with respect to t on (0,x]. Then use (a).) =