(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 tha...
Let {h} be a sequence ofRiemann integrable functions on [a,b], such that for each x, {h(x)) is a decreasing sequence. Suppose n) converges pointwise to a Riemann integrable function f Prove that f(x)dxf(x)dx. lim n00 Let {h} be a sequence ofRiemann integrable functions on [a,b], such that for each x, {h(x)) is a decreasing sequence. Suppose n) converges pointwise to a Riemann integrable function f Prove that f(x)dxf(x)dx. lim n00
12. Let f be integrable on a closed interval [a, b]. Suppose that there is a real number C such that f(x) 2C for all E a, b (1) Prove that if C>0, then 7 is also integrable on la,b] (6 Marks) (2) If C 0, i, still integrable (assuming f(x)关0 for any x E [aM)? If yes, supply a short proof. If no, give a counterexample. (6 Marks) 12. Let f be integrable on a closed interval [a, b]....
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...
(11) Let (,A. /) be a measure space. Let g 2 - R* be a measurable function which is integrable on a set A E A. Let f, : O -> R* be a sequence of measurable functions such that g(x) < fn(x) < fn+1(x), for all E A and n E N. Prove that lim fn d lim fn du noo A (11) Let (,A. /) be a measure space. Let g 2 - R* be a measurable function...
8. Suppose f : la,b] → R is monotone increasing. Prove that f is integrable. Of course you may not use the theorem that monotone functions are integrable! 8. Suppose f : la,b] → R is monotone increasing. Prove that f is integrable. Of course you may not use the theorem that monotone functions are integrable!
4a. (5 pts) Let f, g: [a, b -R be integrable. Show that la, blR, {f (x),g x)) h h (x) max and k[a, bR, k (x) min {f (x),g (x)) integrable. Hint: Observe that, for all a, b e R, max{a, b}= (a+ b+ la - bl) and min{a, b} (a+b-la -bl). are
Problem 4: Let f: [0, 1] → R be an integrable function that is continuous at 0. Prove that lim f(") dx = f(0). n+Jo [ Hint: there are several approaches. It might help to first show that for a fixed 0 <b< 1, we have limn700 Sº f(x) dx = b. f(0). ]
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...
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...
#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...