a) Let f be integrable on [a, b]. Suppose c ∈Rand g : [a + c, b + c] →Rsuch that g(x) = f (x −c), x ∈[a + c, b + c] Prov
Posted: Thu May 12, 2022 1:44 pm
a) Let f be integrable on [a, b]. Suppose c ∈Rand g : [a + c, b
+ c] →Rsuch that
g(x) = f (x −c), x ∈[a + c, b + c]
Prove that g is integrable on [a + c, b + c] and
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 ∈Rand integrate h(t + y) = h(t) + h(y) with
respect to t on [0, x]. Then use (a).)
Use the hint as much as possible
obtc V f() dr g(2) dar atc
+ c] →Rsuch that
g(x) = f (x −c), x ∈[a + c, b + c]
Prove that g is integrable on [a + c, b + c] and
- A Let F Be Integrable On A B Suppose C Rand G A C B C Rsuch That G X F X C X A C B C Prov 1 (19.35 KiB) Viewed 25 times
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 ∈Rand integrate h(t + y) = h(t) + h(y) with
respect to t on [0, x]. Then use (a).)
Use the hint as much as possible
obtc V f() dr g(2) dar atc