   Chapter 2.5, Problem 72E

Chapter
Section
Textbook Problem

(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 or the statement in part (b) also true? In other word, if | f | is continuous, does it follow that f is continuous? If so, prove it. If not, find a counterexample.

(a).

To determine

To Prove: The absolute value function F(x)=|x| is continuous everywhere.

Explanation

Given:

The function F(x)=|x|

Formula used:

(1) Definition of the limit

Calculation:

First check the limit of the function at x=0

Right hand limit at x=0.

limh0+F(0+h)=limh0+|h|=0

Left hand limit at x=0.

limh0F(0h)=limh0|0h|=0

And F(0)=|0|=0

Since limh0+F(x)=0=limh0F(x)

Therefore, F(x)=|x| is continuous at x=0

(b).

To determine

To prove: The function |f(x)| is continuous on an interval if f(x) is continuous on the same interval.

(c)

To determine

To find: The converse of the part (b) is also true, If not find the counter example.

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