I just asked a specific homework question on this topic, but I want a more general explanation for how to go about proving continuity with this method.
I can't even wrap my head around what the proof is really saying, let alone figure out the steps to prove a specific function is continuous.
Edit: okay, here is a specific example. Maybe a walkthrough would be easier for you guys to help me out:
Prove that $g(x) = \frac{1}{x^2 - 4}$ is continuous on $\mathbb{R} \setminus \{ -2, 2\}$.