Question

Let $f(x)=[x{]}^2-\left[x^2\right],x\in [-2,2]$ where [ ] represents greatest integer function. Let $m,k$ represent the number of irrational values of $x$ at which $f$ is not differentiable and the number of integral values of $x$ at which $f$ is continuous respectively. Find $m-k$.

Answer 1

Answer: 3

$f(x)=[x{]}^2-\left[x^2\right]$
$=2-2=0,x=-2$
$=4-3=1,-2<x\le -\sqrt{3}$
$=4-2=2,-\sqrt{3}<x\le -\sqrt{2}$
$=4-1=3,-\sqrt{2}<x<-1$
$=1-1=0,x=-1$
$=1-0=1,-1<x<0$
$=0,x=0$
$=0-0=0,0<x<1$
$=1-1=0,x=1$
$=1-1=0,1<x<\sqrt{2}$
$=1-2=-1,\sqrt{2}\le x<\sqrt{3}$
$=1-3=-2,\sqrt{3}\le x<2$
$=4-4=0,x=2$
The number of irrational values of $x$ at which $f$ is not differentiable $=4=m$
The number of integral values of $x$ at which function $f$ is continuous (at $x=1)=1=k$
$\Rightarrow m-k=4-1=3$