$\lim\limits_{x\to 1^-}f(x)=1$
$\lim\limits_{x\to -1^+}=|-1|=1$
$\Rightarrow \lim\limits_{x\to{-1}}f(x)=f(-1)=\lim\limits_{x\to{-1^+}}f(x)$
$f$ is continuous at $x=-1$
It can be shown that it is not differentiable at $x=-1$
Hence (d) is the correct answer.