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: 89


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Work Step by Step

We are given the function: $f(x)=\begin{cases} 1,\text{ for }x \text{ rational}\\ -1,\text{ for }x \text{ irrational} \end{cases}$ Determine $[f(x)]^2$: $[f(x)]^2=\begin{cases} 1^2,\text{ for }x \text{ rational}\\ (-1)^2,\text{ for }x \text{ irrational} \end{cases}$ $[f(x)]^2=\begin{cases} 1,\text{ for }x \text{ rational}\\ 1,\text{ for }x \text{ irrational} \end{cases}$ Let $x_0$ be a rational number. There are infinitely many irrational numbers to the left and to the right of $x_0$, so $f^(x)$ is continuous in $x_0$. The function takes the same value for rational and irrational numbers. Therefore $f^2(x)$ is continuous on $\mathbb{R}$.
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