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: 81


False statement

Work Step by Step

We are given: $f(x)$ discontinuous in $x=c$ $g(x)$ discontinuous in $x=c$ We have to check in $f(x)+g(x)$ is also discontinuous in $c$. Consider the example: $f(x)=\begin{cases} x+1,\text{ for }x\leq 2\\ 2x+1,\text{ for }x>2 \end{cases}$ $g(x)=\begin{cases} -2x+2,\text{ for }x\leq 2\\ -x-2,\text{ for }x>2 \end{cases}$ Compute $f(x)+g(x)$: $f(x)+g(x)=\begin{cases} x+1-2x+2,\text{ for }x\leq 2\\ 2x+1-x-2,\text{ for }x>2 \end{cases}$ $f(x)+g(x)=\begin{cases} -x+3,\text{ for }x\leq 2\\ x-1,\text{ for }x>2 \end{cases}$ The functions $f$ and $g$ have a jump discontinuity in $x=2$. We check if $f+g$ is continuous in $x=2$: $\displaystyle\lim_{x\rightarrow 2^{-}} (f(x)+g(x))=\displaystyle\lim_{x\rightarrow 2^{-}} (-x+3)=1$ $\displaystyle\lim_{x\rightarrow 2^{+}} (f(x)+g(x))=\displaystyle\lim_{x\rightarrow 2^{-}} (x-1)=1$ $(f+g)(2)=-2+3=1$ As the left and right hand limits and the value of $(f_g)$ in $x=2$ are equal, the sum function is continuous in $x=2$. Therefore the statement is FALSE.
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