Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 11 - Section 11.3 - Limits and Continuity - Concept and Vocabulary Check - Page 1160: 6

Answer

The statement “A piecewise function is always discontinuous at one or more numbers” is false.

Work Step by Step

Consider the piecewise function, $ f\left( x \right)=\left\{ \begin{align} & x-1\text{ if }x\le 0 \\ & {{x}^{2}}+x-1\text{ if }x>0 \end{align} \right.$. Since, $ x-1\text{ and }{{x}^{2}}+x-1$ both are continuous polynomial functions. Therefore, the function can have the point of discontinuity only at $ a=0$. Now check the discontinuity of the function at $ a=0$. Find the value of $ f\left( x \right)$ at $ a=0$, $\begin{align} & f\left( 0 \right)=0-1 \\ & =-1 \end{align}$ The function is defined at the point $ a=0$. Now find the value of $\,\underset{x\to 0}{\mathop{\lim }}\,f\left( x \right)$. First check the left hand limit of $\,f\left( x \right)$. $\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,f\left( x \right)=0-1=-1$ Now find the right hand limit of $\,f\left( x \right)$, $\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,f\left( x \right)={{0}^{2}}+0-1=-1$ Thus the left hand limit and right hand limit are equal, that is $\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,f\left( x \right)=-1=\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,f\left( x \right)$ Thus, $\underset{x\to 0}{\mathop{\lim }}\,f\left( x \right)=-1$ From the above steps, $\,\underset{x\to 0}{\mathop{\lim }}\,f\left( x \right)=-1=f\left( 0 \right)$ Thus, the function satisfies all the properties of being continuous. Hence, the function $ f\left( x \right)=\left\{ \begin{align} & x-1\text{ if }x\le 0 \\ & {{x}^{2}}+x-1\text{ if }x>0 \end{align} \right.$ is continuous at $0$. Thus, the function $ f\left( x \right)$ is not discontinuous for any number. Hence, the statement is false.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.