The formal definition of continuity is a local property (the definition of continuity at a point is a property of the germ of the function at this point). Why is it a good decision to make the definition local?
To motivate my question: The starting intuition behind continuity for a real valued function $f:D\to\mathbb R$ is often:
$f$ is continuous when its (whole) graph can be drawn without lifting the pen.
This intuition suggests that the definition should be global in a sense that it takes the whole function into account. This intuition does also not explain, why we should define continuity at a point. So:
What is the benefit of having a local definition of continuity?
Note: This is a follow-up question of How can I motivate the formal definition of continuity? Although both questions are highly connected, I think the answers to both questions will be a different. That's why I made a new question. I hope that's okay...