You're right: It only makes sense to say that a function $f$ is continuous in $x$ if $x$ is in the domain of $f$, i.e. if $f$ is defined in $x$.
Let me add that for other examples it could be possible that $f$ can be extended to a continuous function
$$
\tilde{f}(x) = \begin{cases} f(x), &x\neq 0,\\ c, & x=0,\end{cases}
$$
with some value $c\in\mathbb{R}$, which is then a function defined on the larger domain $[-1,1]$ including $x=0$ (one says that $f$ possesses a removable discontinuity in $0$ if that works).
You only obtain a continuous function by that process if $c = \lim_{x\nearrow 0} f(x) = \lim_{x\searrow 0} f(x)$ and this is obviously not the case in your example, since $\lim_{x\nearrow 0} f(x) = -\infty \neq \infty = \lim_{x\searrow 0} f(x)$.