4.2.2Please include a formal proof. Definition 4.2.1. Let I CR be an open interval, let c E I and let f:1¬R be afunction.1. The function f is differentiable at e iflim f(x) – f(c)x-cexists; if this limit exists, it is called the derivative of f at c, and it is denotedf'(c).2. The function f is differentiable if it is differentiable at every number in 1. Iff is differentiable, the derivative of f is the function f' : I → R whose valueat x is f'(x) for all x €I.Lemma 4.2.2. Let ICR be an open interval, let c e I and let f : 1¬R be a function.Then f is differentiable at e if and only iflimexists, and if this limit exists it equals f'(c).Proof. Suppose that f is differentiable at c. Let F:1- {c} →R be defined byF(x) = L1) – f(c)X-efor all x €I- {c}. Then lim F(x) exists and equals f'(c).Let J= {x-c|x€ I}, and let g: J – {0} → I – {c} be defined by g(x) = x+cfor all x €J- {0}. By Exercise 3.2.1 and Exercise 3.2.5 we know that lim g(h) =c.It is struightforward to verify that(Fog)(h) = Lc+h) – (c)for all h EJ-{0}. Because lim F(x) exists, then Theorem 3.2.12 implies that lim (Fo8)(h) exists and lim (Fog)(h) = lim F(x), which is equivalent to saying thatS(c+h) – {(c)limexists and equals S'(c).The other implication is similar, and we omit the details. Exercise 4.2.2. Let f: R → R be defined byif x E Qf(x) =|x², ifx€R- Q.Using only the definition of derivatives and Lemma 4.2.2, determine whether f isdifferentiable at 0. If it is, find f'(0); if it is not, show why not.

Answered: Image /qna-images/answer/8b349e0b-0263-4887-a684-d2b80f5fe4b5.jpg

Answered: Sx, if xe Q if x ER – Q. f(x) = Using…

Sx, if xe Q if x ER – Q. f(x) = Using only the definition of derivatives and Lemma 4.2.2, determine whether f is differentiable at 0. If it is, find f'(0); if it is not, show why not.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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