Let $[x]$ denote the greatest integer less than or equal to x. If $f(x)=[x\sin\pi x]$, then $f(x)$ is

Question

$\begin{array}{1 1}(a)\;\text{continuous at x=0}\\(b)\;\text{continuous in (0,0)}\\(c)\;\text{differentiable at x=1}\\(d)\;\text{differentiable in (-1,1)}\end{array}$