Making sense of continuity at a point where f(x) = Infinity?

[AxiomOfChoice]

Is there a way to make sense of the following statement: "$f$ is continuous at a point $x_0$ such that $f(x_0) = \infty$?" The standard definition of continuity seems to break down here: For any $\epsilon > 0$, there is no way to make $|f(x_0) - f(x)| < \epsilon$, since this is equivalent to making $|\infty - f(x)| < \epsilon$, which cannot happen, since $\infty - y = \infty$ for every $y\in \mathbb R$ and $\infty - \infty$ is undefined. So is there any way to make sense of continuity of an extended real-valued function at a point where it's infinite?

 

[Mute]

Could you try regarding $f(x_0) = \infty$ as $1/f(x_0) = 0$, and so make

$$\left|\frac{1}{f(x_0)}-\frac{1}{f(x)} \right| < \epsilon$$
?