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. $x=2$
• D. $x=1$

Answer 1

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

The correct options are
A $x=-1$
C $x=1$
D $x=2$

Let $x=n$ be any integer not equal to zero. Then

${lim}_{x\to n^{-}}x\cos \left(\pi (x+\left[x\right])\right)$

$=n\cos \left(\pi (n+n-1)\right)$

$=n\cos (2n-1)\pi =-n$

${lim}_{x\to n^{+}}x\cos \left(\pi (x+\left[x\right])\right)$

$=n\cos \left(\pi (n+n)\right)$

$=n\cos 2n\pi =n$

LHL$\ne$RHL

$\Rightarrow$Limit does not exist at any non-zero integer $n$

$\therefore f$ is discontinuous at $x=-1,1,2$

At $x=0,$LHL$=$RHL$=0=f\left(0\right)$

$\therefore f$ is continuous at $x=0$