hope this will help you
8. Suppose f : la,b] → R is monotone increasing. Prove that f is integrable. Of course you may no...
#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...
(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
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]....
31 (a) If fis integrable, prove that fa is integrable. Hint: Given e>0, let h and k be step functions such that h f k and j (k-h) < ε/M, where M is the maximum value of Ik(x) +h(x)]. Then prove that h and k2 are step functions with h' srsk (we may assume that OShSSk since f is integrable if and only if I is-why?), and that I (k2 - h2) <e. Then apply Theorem 3.3. (b) If fand...
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
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...
Integral: If you know all about it you should be easy to prove..... Let f:[a,b]→R and g:[a,b]→R be two bounded functions. Suppose f≤g on [a,b]. Use the information to prove thatL(f)≤L(g)andU(f)≤U(g). Information: g : [0, 1] —> R be defined by if x=0, g(x)=1; if x=m/n (m and n are positive integer with no common factor), g(x)=1/n; if x doesn't belong to rational number, g(x)=0 g is discontinuous at every rational number in[0,1]. g is Riemann integrable on [0,1] based...
Three questions!please! 7. Prove that J(x) is integrable on (0,b), and calculate their integral. 8. Prove that the following function is integrable on [0, 1], and calculate the integral. 1 if for some n E N 0 (z)= otherwise. 8. Prove that if f is integrable on (a, b, then f2 is also integrable on la,b
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
Suppose that is integrable on [a,b]. → R is positive and integrable. Show that, f f(x) : [a,