4a. (5 pts) Let f, g: [a, b -R be integrable. Show that la, blR, {f...
3. Let f, g : a, bl → R be functions such that f is integrable, g is continuous. and g(x) >0 for al x E [a, b]. Since both f,g are bounded, let K> 0 be such that f(x)| 〈 K and g(x)-K for all x E la,b] (a) Let η 〉 0 be given. Prove that there is a partition P of a,b] such that for all i (b) Let P be a partition as in (a). Prove...
3. Let f, g : [a,b] → R be functions such that f is integrable, g is continuous, and g(x) >0 for all r E [a, b] Since both f,g are bounded, let K >0 be such that lf(z)| K and g(x) K for all x E [a3] (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that U (P. f) _ L(P./) < η and Mi(P4)-mi(P4) < η for all...
3. Let f, g : a, b] → R be functions such that f is integrable, g is continuous. and g(x) 〉 0 for all x є a,b]. Since both f, g are bounded, let K 〉 0 be such that |f(x) K and g(x) < K for all x E [a,b (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that for all i 2. (b) Let P be a...
Problem 10. Let f,g: [a,b] -R be Riemann integrable functions such that f(x) < g(x) for all x E [a,b]. Prove that g(x)
(6) Let a<b, and suppose the function f is integrable a, b. Show that for every infinite on IR such that g(x)= f (x) for all e [a,b]\ S subset SC [a, b), there is a function g: [a, b and g is not integrable. [ef: 7.1.3 in text. (7) Show directly that if the function f : [a,b possibly at one point o (a,b), thenf is integrable on fa, b). R is continuous everywhere in a, b) except (6)...
(6) Let a<b, and suppose the function f is integrable a, b. Show that for every infinite on IR such that g(x)= f (x) for all e [a,b]\ S subset SC [a, b), there is a function g: [a, b and g is not integrable. [ef: 7.1.3 in text. (7) Show directly that if the function f : [a,b possibly at one point o (a,b), thenf is integrable on fa, b). R is continuous everywhere in a, b) except (6)...
(c) Let f :la,b- R be an integrable function. Prove that lim . (Your argument should include why faf makes sense for a < x < b.) (c) Let f :la,b- R be an integrable function. Prove that lim . (Your argument should include why faf makes sense for a
analysis 2 III. Let f,g be Riemann integrable on [a, b). Show that, for any k>0, f (x)g(x)dr. IV. Show that eb 6 III. Let f,g be Riemann integrable on [a, b). Show that, for any k>0, f (x)g(x)dr. IV. Show that eb 6
Problem 1. Consider the function f(x)- 3.12 show that f is Riemann integrable on [0.2] and use the definition to find .后f(x)dr Problem 2. Consider the function -2, zEQ 2, O f(r) = Show that f is not Riemann integrable on 0,1 but s Reemann integrable on this interval. Problem 3. (a) Let f be a real-valued function on a, b] such thatf()0 for all c, where c E [a, b Prove that f is Riemann integrable on a, b...
#4 (4) Use the Box-sum criterion to prove that if f is integrable on [a, b] and is also integrable on |b,e, then f is integrable on la, e) and Je fdr- o fdz+ (5) Suppose that (r) 2 0 and is continuous on a, b). Prove that if f - 0, then f(x) = 0 for all x E a,b]. Hint: Assume to the contrary that there is some r E [a, b] where f(x) > 0. What can...