Function $g(x) = \lvert x\rvert$, $x \neq 0$: $g(0)$ not defined?

Question

I found this statement in topic limits:

Consider the following function $g(x) = \lvert x\rvert$, $x \neq 0$. Observe that $g(0)$ is not defined.

How come $g(0)$ not defined when the absolute value function is defined at $0$?

Answer 1

In order to define a function, you need three things:

1. a domain,
2. a codomain, and
3. a way of mapping things in the domain to things in the codomain.

In this case, $x=0$ is explicitly excluded from the domain of $g$, hence $g(0)$ is explicitly not defined. That being said, there is another function, call it $f :\mathbb{R}\to\mathbb{R}$, defined by $f(x) = |x|$ which is defined at zero, and is equal to $g$ everywhere where both $f$ and $g$ are defined. However, note that $f$ and $g$ are not the same function---they are different functions as they have different domains.

Moreover, as noted in the comments above, we could define yet another function $h:\mathbb{R}\to \mathbb{R}$ by setting $$ h(x) := \begin{cases} |x| & \text{if $x\ne 0$, and} \\ 7 & \text{if $x=0$.} \\ \end{cases} $$ This function will have the same domain and codomain as $f$, but is a different function from $f$, as it maps the domain to the codomain differently (i.e. $f(0) = 0 \ne 7 = h(0)$).