Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points,…

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 :
  1. 6
  2. 8
  3. 9
  4. 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.
$\therefore \quad m+n=8$

Asked in: JEE Main 2025 (24 Jan Shift 2)

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