Question

(a) Show that the absolute value function $ F(x) = | x | $ is continuous everywhere. (b) Prove that if $ f $ is a continuous function on an interval, then so is $ | f | $. (c) Is the converse of the statement in part (b) also true? In other words, if $ | f | $ is continuous, does it follow that $ f $ is continuous? If so, prove it. If not, find a counterexample.

Name: a show that the absolute value function fx x is continuous everywhere b prove that if f is a continuous function on an interval then so is f c is the converse of the statement in part b also true in 2
Uploaded: 2021-08-26T00:16:27-08:00
Duration: 4 min
Channel: Jimmy Yao
Description: a show that the absolute value function fx x is continuous everywhere b prove that if f is a continuous function on an interval then so is f c is the converse of the statement in part b also true in 2

          (a) Show that the absolute value function $ F(x) = | x | $ is continuous everywhere.
(b) Prove that if $ f $ is a continuous function on an interval, then so is $ | f | $.
(c) Is the converse of the statement in part (b) also true? In other words, if $ | f | $ is continuous, does it follow that $ f $ is continuous? If so, prove it. If not, find a counterexample.

Added by Fake F.

Calculus: Early Transcendentals

James Stewart 8th Edition

Chapter 2

Instant Answer

Solved by Expert Jimmy Yao

08/26/2021

Step 1

Step 1: To show that the absolute value function $ F(x) = | x | $ is continuous everywhere, we need to show that $ \lim_{x \to a} | x | = | a | $ for all $ a \in \mathbb{R} $. Show more…

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(a) Show that the absolute value function $ F(x) = | x | $ is continuous everywhere. (b) Prove that if $ f $ is a continuous function on an interval, then so is $ | f | $. (c) Is the converse of the statement in part (b) also true? In other words, if $ | f | $ is continuous, does it follow that $ f $ is continuous? If so, prove it. If not, find a counterexample.