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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).)

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter5: Rings, Integral Domains, And Fields
Section5.2: Integral Domains And Fields
Problem 16E: Prove that if a subring R of an integral domain D contains the unity element of D, then R is an...
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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
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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