   Chapter 2.5, Problem 68E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# For what values of x is g continuous? g ( x ) = { 0   if   x   is rational 1   if   x   is irrational

To determine

To find: The values of x for which the function g(x)={0if xxif x

is continuous

Explanation

Check the continuity of g on x=0

0g(x)x for all x.

So,x<0g(x)x which implies that xg(x)x

Thus,|x|g(x)|x|

Take limits on the above equation

limx0{|x|}limx0g(x)limx0|x| which becomes 0limx0g(x)0

Recall that Squeeze theorem states if for f(x)g(x)h(x) for all x in an open interval around x=a and limxaf(x)=limxah(x)=l, then limxag(x)=l

So, limx0g(x)=0

Thus, limx0g(x)=g(0)

The function g(x)={

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