The value of k which makes

(d) none of these

If $f(x)$ is continuous at $x=0$, then

$\lim _{x \rightarrow 0} f(x)=f(0)$

$\Rightarrow \lim _{x \rightarrow 0}\left(\sin \frac{1}{x}\right)=k$

But $\lim _{x \rightarrow 0}\left(\sin \frac{1}{x}\right)$ does not exist. Thus, there does not exist any $k$ that makes $f(x)$ a continuous function.