Let [x] be the greatest integer less than or equals to x . Then, at...

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

$⇒ f\left(x\right) = {\left(- 1\right)}^{\left[x\right]}xcos\left(\pix\right)$

$[ x ]$ is discontinuous at all integers and $xcos\pix$ is continuous everywhere.

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

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