Please check the correct answers. f(0) = L? at 2. (2 points) Let a, L €...
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem) 2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
Please give clear detailed explanation. Let a 0 and suppose that the function f is Riemann integrable on [0, a]. Prove that f(a-x) dx = 2S0[f(x) + f(a- 1 ca f(x) dx = x)j dx. Prove that f' in(1 + tan(a) tan(x)) dx = a ln(sec(a)) (0<a<T/2) Let f: [0, 1] → R be defined by f(x) = VX , 0 1 , and let x 2 n-1,2 be a partition of [0, 1]. Calculate lRll and show that lim...
Definition 1. A function f(x) defined on (-L, L] is called piece-wise continuous if there are finitely many points xo =-L < x1 < x2 < < xn-L such that f is continuous on (xi, i+1) and so that the limits lim f(z) and lim f(x) both crist for each a,. To save space we write lin. f(x) = f(zi-) ェ→z, lim, f(x) = f(zit), ェ→ Sub-problem 5. Let f(x)-x on (-2,-1), f(x) = 1 on (-1,0) and f(x)--z on...
+20 Problem 7. Let f :D + R, xo be an accumulation point of D and assume lim f(x) = L. Use the e-8 definition of the limit (not theorems or results from class or the text) to prove the following: (a) The function f is “bounded near xo”: there is an M ER and a 8 >0 such that for x E D, 0 < l< – xo<8 = \f(x) < M. Hint: compare with the proof that a...
hint This exercise 5 to use the definition of Riemann integral F. Let f : [a, b] → R be a bounded function. Suppose there exist a sequence of partitions {Pk} of [a, b] such that lim (U(Pk, f) – L (Pk,f)) = 0. k20 Show that f is Riemann integrable and that Så f = lim (U(P«, f)) = lim (L (Pk,f)). k- k0 1,0 < x <1 - Suppose f : [-1, 1] → R is defined as...
Problem 4 (4 points each). Let S = R {0}. (a) Let f: S R be f(x) = cos(1/x). Show that lim-0 f(x) does not exist. (b) For any fixed a > 0, let f: S+R be f(x) = rºcos(1/x). Show that lim -- f(x) = 0. (c) Find a value be R for which the function f: R+R given by f(x) = { 2" cos(1/x) if r +0, if x = 0, is continuous at 0. Is this b...
7. (10%) Let f: [0,1] R be defined by _x xe[0,1]n f(x) 0 otherwise Is fe L[0,1]? If yes, find its Lebesgue integral. i) Is feR[O,1] ? If yes, find its Riemann integral. ii) ii) What is lim || |, ? 7. (10%) Let f: [0,1] R be defined by _x xe[0,1]n f(x) 0 otherwise Is fe L[0,1]? If yes, find its Lebesgue integral. i) Is feR[O,1] ? If yes, find its Riemann integral. ii) ii) What is lim ||...
Let fi and f2 be functions such that lim e s f1 (2) = + and such that the limit L2 = lim a s f2 (x) exists. Which one of the following is NOT correct? O limas (f1f2)(x) = 0 if L2 = 0. limas (fi + f2)(x) = too if L2 = -0. Olim as (f1f2) (x) = too if 0 <L2 5+co. lim a s (f1f2)(x) = - it L2 = -. Which one of the following...
Answer both and show work. 1y=fle) L-5. Given the graph of the function f(x) shown to the right-the same graph is - used for L-2. Identify any points where f(2) is not con- tinuous and explain why using precise mathematical statements in relation to how the definition of continuity is not sat- isfied. You may refer to work in L-2 rather than repeat it. L-3. Given the function g(2) defined piecewise below, determine each of the following values, showing clearly...
Real analysis 10 11 12 13 please (r 2 4.1 Limit of Function 129 se f: E → R, p is a limit point of E, and limf(x)-L. Prove that lim)ILI. h If, in addition, )o for all x E E, prove that lim b. Prove that lim (f(x))"-L" for each n E N. ethe limit theorems, examples, and previous exercises to find each of the following limits. State which theo- rems, examples, or exercises are used in each case....