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)

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

$x=0^{-}\to f\left(0^{-}\right)=0x=0^{+}\to f\left(0^{+}\right)=0$
Continuous at $x=0$
$x=1^{-}\to f\left(1^{-}\right)=1\cos \left(\pi \right(1+0\left)\right)=-1x=1^{+}\to f\left(1^{+}\right)=1\cos \left(\pi \right(1+1\left)\right)=1$
Discontinuous at $x=1$
Similarly discontinuous at $x=-1,2$