Let h: RR be integrable on every bounded interval and h(x + y)=h(x) +h(y) for any r, y ER Show that h(x) = cr for any r ER, 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, r]. Then use (a).)
Let h: RR be integrable on every bounded interval and h(x + y)=h(x) +h(y) for any r, y ER Show that h(x) = cr for any r ER, 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, r]. Then use (a).)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Please solve b part only
![Let f be integrable on [a, b]. Suppose c ER 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
cb+c
[ f(x) dx = fost
9(2) d
a+c
(b) (
Let h: R→ R be integrable on every bounded interval and
h(x+y)=h(x) +h(y) for any r, y ER
Show that h(x) = cx for any r R, 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, z]. Then use
(a).)
4. (a) (1
dx](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F444a28ed-437f-4929-a1ec-5907998afa93%2F169f283d-4235-4b7d-bd0b-9b5a78117882%2Figshuwj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let f be integrable on [a, b]. Suppose c ER 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
cb+c
[ f(x) dx = fost
9(2) d
a+c
(b) (
Let h: R→ R be integrable on every bounded interval and
h(x+y)=h(x) +h(y) for any r, y ER
Show that h(x) = cx for any r R, 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, z]. Then use
(a).)
4. (a) (1
dx
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