Let f : ( - 2 , 2 ) R be defined by f ( x ) = { x [ x ] , - 2 x 0 ( x - 1 ) [ x ] , 0 x 2 where [ x ] denotes the greatest integer function. If m and n respectively are the number of points in (- 2, 2) at which y = | f ( x ) | is not conti

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Solution

Q.
Let $f : (- 2, 2) \to R$ be defined by $f (x) =$ ${\begin{matrix} x [x], & - 2 < x < 0 \\ (x - 1) [x], & 0 \leq x < 2 \end{matrix}$ where $[x]$ denotes the greatest integer function. If $m$ and $n$ respectively are the number of points in (- 2, 2) at which $y =$ $| f (x) |$ is not continuous and not differentiable, then $m + n$ is equal to _____. [2023]

Ans.
(4)

$f (x) or$ $| f (x) |$ $=$ ${\begin{matrix} - 2 x, & - 2 < x < - 1 \\ - x, & - 1 \leq x < 0 \\ 0, & 0 \leq x \leq 1 \\ x - 1, & 1 \leq x < 2 \end{matrix}$

$Point of discontinuity: m = - 1 i.e., one in number$ $Point of non-differentiability: n = - 1, 0, 1 i.e., 3 in number.$

$∴$ $m + n = 1 + 3 = 4$

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