Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x)=[x]+|x-2|,-2 \lt x \lt 3$, is not continuous and not differentiable. Then $\mathrm{m}+\mathrm{n}$ is equal to :

• A

  6
• Correct answer
  B

  8
• C

  9
• D

  7

Solution

$\begin{aligned} & f(x)=[x]+|x-2|,-2 < x < 3 \\ & \therefore f(x)=\left\{\begin{array}{l} -x,-2 < x < -1 \\ 1-x,-1 \leq x < 0 \\ 2-x, 0 \leq x < 1 \\ 3-x, 1 \leq x < 2 \\ x, 2 \leq x < 3 \end{array}\right. \end{aligned}$
It is clearly discontinues at 4 points and nondifferentiable at 4 points. $\therefore \quad m+n=8$

Hence, the correct answer is option: 'B'.