Question

Let $\left[x\right]$ be the greatest integer less than or equal to $x$. Then at which of the following point(s) the function $f\left(x\right)=x\cos \left(\pi \right(x+\left[x\right]\left)\right)$ is discontinuous

• A. $x=-1$
• B. $x=0$
• C. $x=1$
• D. $x=2$

Answer 1

Answer: (A), (C), (D)

$x=2$

$f\left(x\right)=x\cos \left(\pi \right(x+\left[x\right]\left)\right)$
At $x=0,$
$L.H.L.=f\left(0^{-}\right)=0R.H.L.=f\left(0^{+}\right)=0$
So, $f\left(x\right)$ is continuous at $x=0$
At $x=1,$
$L.H.L.=f\left(1^{-}\right)=1\cos \left(\pi \right(1+0\left)\right)=-1R.H.L.=f\left(1^{+}\right)=1\cos \left(\pi \right(1+1\left)\right)=1$
So, $f\left(x\right)$ is discontinuous at $x=1$
At $x=-1,$
$L.H.L.=f(-1^{-})=1$
$R.H.L.=f(-1^{+})=-1$
So, $f\left(x\right)$ is discontinuous at $x=-1$
At $x=2,$
$L.H.L.=f\left(2^{-}\right)=-2$
$R.H.L.=f\left(2^{+}\right)=2$
So, $f\left(x\right)$ is discontinuous at $x=2$