Question

The set of points where the function $f\left(x\right)=\left[x\right]+|1-x|,-1\le x\le 3$, where $[.]$ denotes the greatest integer function, is not differentiable, is

• A. $\{-1,0,1,2,3\}$
• B. $\{-1,0,2\}$
• C. $\{0,1,2,3\}$
• D. $\{-1,0,1,2\}$

Answer 1

Answer: (C)

$\{0,1,2,3\}$

We have
$f\left(x\right)=\{\begin{array}{c}-x,-1\le x<0\\ 1-x,0\le x<1\\ x,1\le x<2\\ 1+x,2\le x<3\\ 5,x=3\end{array}$
Clearly, $f\left(x\right)$ is discontinuous at $x=0,1,2$ and $3$. So, it is not differentiable at these points.
At $x=-1$, we have

${lim}_{x\to -1^{}}f\left(x\right)={lim}_{x\to -1^{}}-x=1=f(-1)$

so, it is continuous at $x=-1$

Also, RHD at $x=-1=-1$ (a finite number)
Therefore, $f\left(x\right)$ is differentiable at $x=-1$