Zero-function
Definition
In the Real Space
$0 : \mathbb{R} \to \mathbb{R}$ defined as follows is called the zero function.
$$ 0(x) = 0 \quad \text{for all } x \in \mathbb{R} $$
In the Vector Space
Let the zero vector of the vector space $V$ be denoted by $\mathbf{0}_{V}$. The zero function $\mathbf{0} : V \to V$ defined on $V$ is as follows.
$$ \mathbf{0}(\mathbf{v}) = \mathbf{0}_{V} \quad \text{for all } \mathbf{v} \in V $$
Explanation
Since the function values are the same for all variables, it is a type of constant function. It is the only function that is both an odd function and an even function.
The zero function itself becomes the zero vector of the function space vector space.