JEE Main 2020 (Online) 5th September Morning ShiftQuestion 1
Let $f(x)=x\cdot\left[\frac x2\right]$, for $-10<x<10$, where $[t]$ denotes the greatest integer function. Then the number of points of discontinuity of $f$ is equal to ________.
Sets, Relations and Function
Answer$8$
Solution
$f(x)=x[x/2]$ on $(-10,10)$. $[x/2]$ jumps at even integers $x=2m$. At such a point the jump of $f$ is $x\cdot(\pm1)=2m$, nonzero unless $x=0$. At $x=0$: $f\to0$ from both sides and $f(0)=0$, so continuous. Even integers in $(-10,10)$ other than 0: $\pm2,\pm4,\pm6,\pm8$, i.e. $8$ points.