Prove the following variant of Theorem 4.14. Suppose f : [a, b] - R is 1-1....
1) Suppose f (a, b) R is continuous. The Carathéodory Theorem says that f(x) is differentiable at -cE (a, b) if 3 (a, b)-R which is continuous, and so that, (a) Show, for any constant a and continuous function (x), that af(x) is continuous at z-c by finding a Carathéodory function Paf(x). (b) Show, for any constants a, B, that if g : (a, b) -R is differentiable at c, with Carathéodory function pg(z), then the linear combination of functions,...
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) =...
2. Rolle's theorem states that if F : [a, b] → R is a continuous function, differentiable on Ja, bl, and F(a) = F(b) then there exists a cela, b[ such that F"(c) = 0. (a) Suppose g : [a, b] → R is a continuous function, differentiable on ja, bl, with the property that (c) +0 for all cela, b[. Using Rolle's theorem, show that g(a) + g(b). [6 Marks] (b) Now, with g still as in part (a),...
Exercise 5.2.4: Prove the mean value theorem for integrals. That is, prove that if f: [a,b]R is continuous then there exists a ce [a,b] such that f = f(e) (b-a)
R i 11. Prove the statement by justifying the following steps. Theorem: Suppose f: D continuous on a compact set D. Then f is uniformly continuous on D. (a) Suppose that f is not uniformly continuous on D. Then there exists an for every n EN there exists xn and > 0 such that yn in D with la ,-ynl < 1/n and If(xn)-f(yn)12 E. (b) Apply 4.4.7, every bounded sequence has a convergent subsequence, to obtain a convergent subsequence...
3. (a) Suppose f : (a, b) + R is differentiable, and there exists M E R such that If'(x) < M for all x € (a, b). Prove that f is uniformly continuous on (a, b). (b) Let f : [0, 1] → [0, 1] be a continuous function. Prove that there exists a point pe [0, 1] with f(p) = p.
Prove the following Theorem: Theorem. f : X → R is continuous + for any open set U C R, the pre-image f(U) is open in the domain of X (i.e., f(U) = XnV for some open set V C R).
Under is for reference (Mean Value Theorem): Suppose that f: R6 + R is a function with the following two properties: flo) = 0, and at at any point Te R6 and any increment h, || DFOD | E || ||. Show that f(B1)) (-1,1). Comment. You should use the Mean Value Theorem at some point in this problem. An interpretation with more jargon is that if the operator norm of Df is at most 1 at all points, then...
Logic (a) Let f : [a, b] → R be a continuous function. Prover that there exists ce [a, b] such that con la silany - be a criminatoria per le sue in elan 5(e) = So gladde (b) Define F:R+R by F(x) = [** V1+e=i&t. Prove by citing the appropriate theorem(s) that F is differentiable on R, and calculate F'(c). Be sure to justify your reasoning at every stage.
9. Suppose that f : [0,-) + R is differentiable and that the derivative f' : [0,00) + R is also differentiable, with f(0) = f'(0) = 0. Suppose also that [f"(x) < 1 for all € [0, 0). a) Show how the Mean Value Theorem can be used to prove that f(x) <r? for all x € (0,00). b) Show how the Cauchy Generalized MVT can be used to prove a stronger statement: |f(7) < 2 for all 2...