·J (I) < 0 for all such y. (Hint: let g(x)--f(x) and use part (a)) 3....
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...
2. If S:= {1/n - 1/min, me N}, find inf S and sup S. 4. Let S be a nonempty bounded set in R. (a) Let a > 0, and let aS := {as : S ES). Prove that inf(as) = a infs, sup(as) = a sup S. (b) Let b <0 and let b = {bs : S € S}. Prove that inf(bs) = b supS, sup(bs) = b inf S. 6. Let X be a nonempty set and...
Let X, Y be two nonempty sets and let f : X → Y. For a, b X we write a ~ b iff f(a) = f(b). Prove that~is an equivalence relation on X Write lely for the equivalence class of x e X with respect to “~" Express [ely in terms of the function f: Irl, = {re x : f(z') a: b: ?? J. (I d o not want to see ..|x ' = {x"e X : r,...
(5) Let f: [0, 1 R. We say that f is Hölder continuous of order a e (0,1) if \f(x) -- f(y)| . , y sup [0, 1] with 2 # 1£l\c° sup is finite. We define Co ((0, 1]) f: [0, 1] -R: f is Hölder continuous of order a}. = (a) For f,gE C ([0, 1]) define da(f,g) = ||f-9||c«. Prove that da is a well-defined metric Ca((0, 1) (b) Prove that (C ([0, 1]), da) is complete...
Please solve all questions 1. Let 0 : Z/9Z+Z/12Z be the map 6(x + 9Z) = 4.+ 12Z (a) Prove that o is a ring homomorphism. Note: You must first show that o is well-defined (b) Is o injective? explain (c) Is o surjective? explain 2. In Z, let I = (3) and J = (18). Show that the group I/J is isomorphic to the group Z6 but that the ring I/J is not ring-isomorphic to the ring Z6. 3....
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 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...
5. Let f(x) = arctan(In x) for all x >0. A graph of y = f(x) is shown in the figure. (a) Find the formula for the derivative f'(x). Then explain how you can deduce from this formula that f is invertible. (b) Find the formula for f-1(x), the inverse of f. (c) What is the domain and range of f-1? (d) Sketch a graph of the function y=f-1(x). (e) Now determine the value of (F-1)(0) using your results from...
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...
I need to answer 1b 2.5. Let f be a real valued function continuous on a closed, bounded Theorem set S. Then there exist x1,X2 S such that f(x1) S f(x) s f(x2) for all x e S. Proor. We recall that if T E' is bounded and closed, then y, - inf T and sup T are points of T (Example 4, Section 1.4). Let T- fIS. By Theorem 2.4, T is closed and bounded. Take x, such that...