Let f: [1,1]→ R be a continuous function. Prove the following statements: (a) If there is c = [-1, 1] such that f(c)f(−c) < 0, then there is de R such that f(d) = 0. (b) If ƒ([−1, 1]) = (-1, 1), then f is not continuous.

Let f: [1,1]→ R be a continuous function. Prove the following statements: (a) If there is c = [-1, 1] such that f(c)f(−c) < 0, then there is de R such that f(d) = 0. (b) If ƒ([−1, 1]) = (-1, 1), then f is not continuous.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section5.2: Exponential Functions

Problem 58E

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Question

4. Let f: [-1, 1] → R be a continuous function. Prove the following statements:
(a) If there is c € [-1,1] such that f(c)f(−c) < 0, then there is de R such
that f(d) = 0.
(b) If ƒ([-1, 1]) = (-1, 1), then f is not continuous.

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Step 1: Introduction

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Step 2: Proving existence of d such that f(d)=0

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Step 3: Proving f is not continuous for part (b)

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