Q31 / 75 · JEE Main 2025 (7 Apr Shift 1) · Mathematics
The number of points of discontinuity of the function $f(\mathrm{x})=\left[\frac{\mathrm{x}^2}{2}\right]-[\sqrt{\mathrm{x}}], \mathrm{x} \in[0,4]$, where $[\cdot]$ denotes the greatest integer function is ________
Correct answer: 8
Solution
Check for $\left[\frac{x^2}{2}\right]$ and $[\sqrt{x}]$ becomes integers.
$\{0,1, \sqrt{2}, 2, \sqrt{6}, \sqrt{8}, \sqrt{10}, \sqrt{12}, \sqrt{14}, 4\}$
Continuous at $0^{+}$, continuous at $4^{-}$ $\left[\frac{x^2}{2}\right]=[\sqrt{x}]$, occurs at $x=\sqrt{2}$
$\Rightarrow$ Not continuous
$\{0,1, \sqrt{2}, 2, \sqrt{6}, \sqrt{8}, \sqrt{10}, \sqrt{12}, \sqrt{14}, 4\}$
Continuous at $0^{+}$, continuous at $4^{-}$ $\left[\frac{x^2}{2}\right]=[\sqrt{x}]$, occurs at $x=\sqrt{2}$
$\Rightarrow$ Not continuous
Hence, the correct answer is 8.