Problem #3: Prove the following theorem:Theorem 2. If f: AR and c E A is an accumulation point of A, then f is continuous at c if and only iflim f(x) = f(c)8个&for every sequence (xn) in A such that xn → c as n → ∞.

Explained below step by step Explanation:Step 1:Step 2:Step 3: Step 4:

Answered: Problem #3: Prove the following…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.3: Factorization In F [x]

Problem 24E

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Specifically part b
Problem #113
1. Consider the following, recursively defined function C (1) = 1 C(2) = 1 C(n) = C(C(n − 1) + C(n − C (n − 1))) Prove that C(n) = 1 for all n.
3. In this problem we study another approach to show that certain equations have a unique solution on an interval [a, b]. The goal is to first write the equation in the form x = f(x) for some function f defined on [a, b]. (a) Suppose that f is continuous on [a, b] and that a ≤ f(x) < b for all x € [a, b]. Prove that there exists c = (a, b) such that c = f(c). Hint: Consider the function g(x) = x-f(x).
10
5. In future courses, you'll meet sequences of functions. For instance, consider we could define a sequence (f₂) of functions fr : R → R inductively via fo(x) = 1, fn+1(x):= 1+ f* fn (t) dt Compute the functions f₁, f2 and f3. The sequence (fn) should seem familiar if you think back to elementary calculus; why?
4. Prove the following: i. f(t) * g(t) = g(t) * f(t). ii. If f and g are piecewise continuous and of exponential order on [0, 0), then (f * g)(t) is of exponential order on [0, ∞).
n True or false: a function f: R → R is continuous at a point a E Rif and only if there exists a sequence (x₂) such that x₂ → a and f(xn) → ƒ(a).
2. Prove using the e-5 notation that the following functions are continuous at p= 1: (a) g2(r) = 2+7 a+7 2a +7 (b) g2(r) = %3D r+7
Consider the sequence of functions {fn(x)} = {n²x(1 – x)"} for a E R and n = = 1,2, 3.... (a) Sketch on the same axis the graphs of f1(x) to f5(x).
6. Inductively prove that B(0) = 0 for all n. Conclude that B(z) is a Coo function.
(b) A function f: A → R is continuous at a point c € A if for any e > 0, unere exists such that whenever 0 0 but 8< f(x)-f(c)| ≤e+e².

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Problem #3: Prove the following theorem: Theorem 2. If f: AR and c E A is an accumulation point of A, then f is continuous at c if and only if lim f(x) = f(c) 8个& for every sequence (xn) in A such that xn → c as n → ∞.