MCQ
The function, $f (x) = [|x|] -|[x]|$ where $[ x ]$ denotes greatest integer function
- Ais continuous for all positive integers
- Bis discontinuous for all non positive integers
- Chas finite number of elements in its range
- ✓All of the above
✓
Answer
Correct option: D.
All of the above
d
$[ | x | ] - | [x] | =$ $\left[ \begin{gathered} 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\, = \,1 \hfill \\ - 1\,\,\,\,\,\,\,\,\,\, - 1 < x < 0 \hfill \\ 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0 \leqslant x \leqslant 1 \hfill \\ 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1 < x \leqslant 2 \hfill \\ \end{gathered} \right.$
$[ | x | ] - | [x] | =$ $\left[ \begin{gathered} 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\, = \,1 \hfill \\ - 1\,\,\,\,\,\,\,\,\,\, - 1 < x < 0 \hfill \\ 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0 \leqslant x \leqslant 1 \hfill \\ 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1 < x \leqslant 2 \hfill \\ \end{gathered} \right.$
$\Rightarrow$ range is $\{0, -1\}$ The graph is 
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