What can we say about undefined values when we encounter them? Is it possible to have a variable with an undefined value? Or do we perhaps just say it is meaningless to talk about $f$ at a point $a$ if $f(a)$ is undefined?

If I have a partial function $f$ for which $f(x)$ is undefined at a particular $x$, we can limit the domain, but still this doesn't help us explain whether we can talk about $f(x)$ at this point, Can we somehow limit the variable space such that $x$ cannot have a value like this?

Is there any concept of equality for undefined values, for example does $undefined = undefined$?

Is it that we would say 'when $x=0$, $y$ has a value that is meaningless, so we cannot talk about either $y$ or $f(x)$?

Say we have some variable $y$ and $y=f(x)$, when $f(x)$ is not defined can $y$ have any value, Or does it just not make any sense to talk about $y$ at all?