JEE Main 2024 (Online) 5th April Evening ShiftQuestion 1
Let $f:[-1,2]\to\mathbf{R}$ be given by $f(x)=2x^{2}+x+\left[x^{2}\right]-[x]$, where $[t]$ denotes the greatest integer less than or equal to $t$. The number of points, where $f$ is not continuous, is :
Limits & Continuity
Solution
Jumps can occur where $x^2$ or $x$ is an integer: $-1,0,1,\sqrt2,\sqrt3,2$. At $x=1$ the jumps of $[x^2]$ and $[x]$ cancel, and at $x=2$ the left limit $8+2+3-1=12$ equals $f(2)$. $f$ is discontinuous at $-1,0,\sqrt2,\sqrt3$: $4$ points.