Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 1 - Functions and Limits - 1.8 Continuity - 1.8 Exercises - Page 94: 67

Answer

$f(x)$ is not continuous for any value of $x$.

Work Step by Step

\[F(x)=\left\{\begin{array}{ll}f(x)\;\;\;\;\;\; ,if \; x\; is \; rational\\ g(x)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;,if \; x\; is \; irrational\end{array}\right. \] $\Rightarrow F(x)$ is continuous at those values of $x$ for which $f(x)=g(x)$. Here, \[f(x)=\left\{\begin{array}{ll}0\;\;\;\;\;\; ,if \; x\; is \; rational\\ 1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;,if \; x\; is \; irrational\end{array}\right. \] $\Rightarrow f(x)$ is continuous at those values of $x$ for which $1=0$ Which is not possible. Hence $f(x)$ is not continuous for any value of $x$.
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