Answer
The zero vector of the vector space $C(-\infty, \infty)$ is the zero function $0(x)$.
Work Step by Step
Since the zero vector, $\textbf 0$, has the property that $$u+\textbf{ 0}=u,$$
then the zero vector of the vector space $C(-\infty, \infty)$ can be calculated as follows;
Let $f_0(x)$ be the zero vector of $C(-\infty, \infty)$, then for any vector $f(x) \in C(-\infty, \infty)$, we have
$$f_0(x)+f(x)=f(x).$$
By solving the above equation we find that $f_0(x)=0=0(x)$, that is, the zero vector is the zero function $0(x)$.
