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 66: 4


$g$ is left-continuous at $c_1=1$ $g(1)=3$

Work Step by Step

A function has a jump discontinuity if both one-sided limits exist, but they are not equal. $\displaystyle\lim_{x\rightarrow 1^{-}} g(x)=2$ $\displaystyle\lim_{x\rightarrow 1^{+}} g(x)=3$ As $\displaystyle\lim_{x\rightarrow 1^{-}} g(x)$ and $\displaystyle\lim_{x\rightarrow 1^{+}} g(x)$, but they are not equal, the function $g$ has a discontinuity at point $x=1$. We also have: $\displaystyle\lim_{x\rightarrow 1^{-}} g(x)=g(1)$ therefore $g$ is left-continuous at $c_1=1$. In order to make $g$ right-continuous in $x=c_1=1$, we must have: $\displaystyle\lim_{x\rightarrow 1^{+}} g(x)=g(1)$ $3=g(1)$ $g(c_1)=g(1)=3$
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