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
?