Let x be the greatest integer less than or equals to x . Then, at which of the following point(s) the…

$f\left(x\right)=x\cos \left(\pi x+\left[x\right]\pi \right)$

$\Rightarrow f\left(x\right)={\left(-1\right)}^{\left[x\right]}x\cos \left(\pi x\right)$

$\left[x\right]$ is discontinuous at all integers and $x\cos \pi x$ is continuous everywhere.

At $x=0$, $f\left(x\right)=0$, so it's continuous.

At other integral points, $f\left(x\right)$ is discontinuous.