TheoremLet f be a continuous function on an open subset U of aBanach space X into a Banach space Y. Let a and b by any distinct points in U suchthat the segment [a, b] is contained in U and f be differentiable at each point of[a, b]. Let g be any continuous linear function on X into Y. Then|| ƒ (b)-f(a) - g (b-a) || ≤c|| b-all,where c is a real number such that|| Df (x) - g || ≤c, for all x in [a, b].

Answered: Image /qna-images/answer/f67c3ff9-b964-48e3-ad2c-c956f1cfdde5.jpg

Answered: Theorem Let fbe a continuous function…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 22E: A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which...

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27. Let , where and are nonempty. Prove that has the property that for every subset of if and only if is one-to-one. (Compare with Exercise 15 b.). 15. b. For the mapping , show that if , then .
A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which of the relations in Exercise 2 areasymmetric? In each of the following parts, a relation R is defined on the set of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if x=2y. b. xRy if and only if x=y. c. xRy if and only if y=xk for some k in . d. xRy if and only if xy. e. xRy if and only if xy. f. xRy if and only if x=|y|. g. xRy if and only if |x||y+1|. h. xRy if and only if xy i. xRy if and only if xy j. xRy if and only if |xy|=1. k. xRy if and only if |xy|1.
If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local extremum offon (a,c) ?
Find mappings f,g and h of a set A into itself such that fg=hg and fh. Find mappings f,g and h of a set A into itself such that fg=fh and gh.
Let (X1, d1) and (X2, d2) be metric spaces and f : X1 -> X2 be a continuous function. Prove that if a is an adherence point of A ⊆ X, then f(a) is an adherence point of f(A). Thank you
Let f : [a, b] → R be a continuous function on the closed interval [a, b] and differentiable on the open interval (a, b). Suppose that c ∈ (a, b) is such that inf{f'(x) : x ∈ (a, b)} < f'(c) < sup{f'(x) : x ∈ (a, b)}. Show that there exist x1, x2 ∈ (a, b) such that f'(c) =(f(x2)− f(x1))/(x2-x1)
let (X,T) be a topological space. Then a function f is continuous at x0 element of X if and only if f is both lower semi continuous and upper semi continuous at x0 element of X.
If the function f is continuous on the closed interval [a, b], then there exists at least one point c element of (a, b) such that
This is a real analysis question. Suppose that (X,d) and (Y,ρ) are metric spaces. (i)The function f:X→Y is continuous if f^(−1) (E) ⊆ X is open whenever E ⊆Y is open. (ii) The function f : X → Y is continuous if for every x ∈ X and for every ε > 0, there exists a δ > 0 such that y ∈ X with d(x,y) < δ implies ρ(f(x),f(y)) < ε. The goal of this problem is to prove a third equivalent characterization. (iii) The function f : X → Y is continuous if for every x ∈ X and for every sequence {xn, n ∈ N} ⊆ X that converges to x, the sequence {f(xn), n ∈ N} ⊆ Y converges to f(x). That is, for every x ∈ X, if lim n→∞ d(xn, x) = 0, then lim n→∞ ρ(f(xn), f(x)) = 0. (a) Prove that (iii) implies (ii). (b) Prove that (i) implies (iii). .
Prove the Theorem 5.5.7 - Let (X1, d1) and (X2, d2) be metric spaces and let f : X1 → X2. Then f is continuous on X1 iff f-1 (G) is an open set in X1 whenever G is an open set in X2.
(a) Assume that (a, b) is a bounded open interval in R. Using Mean Value Theorem, prove that if f : (a, b) → R is an unbounded function (i.e., for every M > 0 there exists x ∈ (a, b) such that | f (x)| > M), then f ′ is also an unbounded function. (b) Give an example of a function f : (a, b) → R, where (a, b) is a bounded open real interval, such that f is differentiable on (a, b), f ′ is unbounded on (a, b), but f is bounded function on (a, b).
Let (+) and (Y ,A) be two topological spaces. Let # be a base for A. Prove that a function f :(4’.r)——({Y ,A) is continuous if and only if the inverse image under ¢ of every member of / is a r- open Set

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