Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 2 - Limits - 2.4 Limits and Continuity - Exercises - Page 68: 88


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Work Step by Step

We are given the function: $f(x)=\begin{cases} x,\text{ for }x \text{ rational}\\ -x,\text{ for }x \text{ irrational} \end{cases}$ First we proved that $f$ is not continuous at $x_0\not=0$. We take a sequence of rational numbers $x_n$ converging to $x_0$ and a sequence of irrational numbers $y_n$ converging to $x_0$. We see that $\displaystyle\lim_{x\rightarrow x_0} f(x)$ does not exist. Then we prove that $f(x)$ is continuous in $x=0$. $f(0)=0$ $-|x|\leq f(x)\leq |x|$ for all $x$. $\displaystyle\lim_{x\rightarrow 0} (-|x|)=\displaystyle\lim_{x\rightarrow 0} |x|=0$ From the pinching theorem we get: $\Rightarrow \displaystyle\lim_{x\rightarrow 0} f(x)=0$ This means that $f$ is continuous only in $x=0$.
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