Question

$f\left(x\right)=\{\begin{array}{c}-1,x<-1\\ -x,-1\le x\le 1\\ 1,x>1\end{array}$ is continuous-

• A. at $x=1$ but not at $x=-1$
• B. at $x=-1$ but not at $x=1$
• C. at both $x=1$ and $x=-1$
• D. at none of $x=1$ and $x=-1$

Answer 1

The correct option is D at none of $x=1$ and $x=-1$

Given $f\left(x\right)=\{\begin{array}{c}-1,x<-1\\ -x,-1\le x\le 1\\ 1,x>1\end{array}$
We will check the continuity at $x=-1$
$f(-1)=-(-1)=1$
$LHL=\underset{x\to -1^{-}}{lim}f\left(x\right)=-1$
$RHL=\underset{x\to -1^{+}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(-1+h)=\underset{h\to 0}{lim}-(-1+h)=1$
Here, $LHL=f(-1)\ne RHL$
Hence, $f\left(x\right)$ is not continuous at $x=-1$
Now,we will check the continuity at $x=1$
$f\left(1\right)=-\left(1\right)=-1$
$LHL=\underset{x\to 1^{-}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(1-h)=\underset{h\to 0}{lim}-(1-h)=-1$
$RHL=\underset{x\to 1^{+}}{lim}f\left(x\right)=1$
Here, $LHL=f\left(1\right)\ne RHL$
Hence, $f\left(x\right)$ is not continuous at $x=1$