The question depends upon whether you are considering $\infty$ to be a number, and over what set you are defining your function. Usually, we do not consider $\infty$ a number, and thus the domain would exclude all places where $\delta(x)=\infty$, that is, $C$. However if you are considering $\infty$ to be a number, your answer would include all places from which you are willing to draw $x$, so probably the reals, but possibly any set.
With that in mind, I'm going to go ahead and say that the answer is probably $C$, because $\infty$ is probably not a number.