Answer Happy

(a) Let f be integrable on [a, b]. Suppose c ∈ R and g : [a
+ c, b + c] → R such that
g(x) = f(x − c), x ∈ [a + c, b + c]
Prove that g is integrable on [a + c, b + c] and
∫b a f(x) dx = ∫b+c a+c g(x) dx
(b) Let h : R → R be integrable on every bounded interval
and
h(x + y) = h(x) + h(y) for any x, y ∈ R
Show that h(x) = cx for any x ∈ R, where c = h(1).
(Hint: Fix any x, y ∈ R and integrate h(t + y) = h(t) + h(y)
with respect to t on [0, x]. Then use (a).
(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), x € (a + c,b+c] Prove that g is integrable on (a + c,b+c] and pb+c /* 2) f(x) dx = = g(2) dx a+c (b) (20 marks) Let h: R+R be integrable on every bounded interval and h(x + y) = h(2) +h(y) for any x,y ER Show that h(x) = cx for any r 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).)