Doubt about $χ_{A}$, the characteristic function of the set $A$

Question

The definition of the characteristic function of $A$ is:

$χ_{A}:A→\{0,1\}$
$χ(x) = \begin{cases} 1, & \mbox{if } x\mbox{ ∈A} \\ 0, & \mbox{if } x\mbox{ ∉A} \end{cases}$

I don't understand, how can this function ever have a $0$ value? For every element in the domain $A$, this function has a value of $1$. If an element is not in the domain this function has value of $0$. But how can a function be defined outside of his domain? I think there is a problem with the definition in this book. Another book gives me a different definition:
Let $X$ be a set and $A$ a subset of $X$. Define the characteristic function of $A$ as follows: $χ(x) = \begin{cases} 1, & \mbox{if } x\mbox{ ∈A} \\ 0, & \mbox{if } x\mbox{ ∉A} \end{cases}$

I interpret this a little differently, as a function $χ_{A}:X→\{0,1\}$.

Could you guys help me clarify my ideas a little? Thank you!

Answer 1

$A$ is supposed to be a subset of $X$ and the domain of $\chi_A$ is then $X$, not $A$ ($\chi_A$ is indeed constant on $A$). The second interpretation you found is more correct. Wikipedia agrees.

Answer 2

The first function should be well-defined, however, it is constant with value $1$. My best guess is, that $A\subset X$ is meant and the function is defined as $\chi_A :X\rightarrow \{ 0, 1\}$, with the same mapping rule.