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

$f(x)=[x]^{2}-\left[x^{2}\right]$
$=2-2=0 \,\, , \,\, x=-2 $
$ =4-3=1 \,\, , \,\, -2< x \leq-\sqrt{3}$
$=4-2=2 \,\, , \,\,-\sqrt{3}< x \leq-\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} \leq x< \sqrt{3}$
$=1-3=-2 \,\, , \,\, \sqrt{3} \leq 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 $x=1)=1=k$
$\Rightarrow m-k=4-1=3$