# Why is continuity defined as a local property?

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...

• Note that the intuition 'can be drawn without lifting the pen' becomes unhelpful if your function has a domain that isn't an interval. – Jessica B Feb 20 '16 at 12:47
• The definition in terms of drawing without lifting the pen is local. You can apply it to any neighborhood. – Ben Crowell Feb 20 '16 at 14:44
• @BenCrowell: Note that even this reformulation does not solve all obstacles. Take for example the Thomae's function. Why is it continuous at all irrational points? This question cannot be explained with the intuition that the function's graph can be drawn locally without lifting the pen... – Stephan Kulla Feb 20 '16 at 14:58
• @tampis: The lifting-the-pen definition isn't a rigorous definition, and I wasn't claiming that it was. – Ben Crowell Feb 20 '16 at 20:24
• The lifting a pen "definition" isn't really even a non-rigorous definition. It's a non-rigorous statement of the theorem that the image, under a continuous function, of a path connected space is path connected. – Adam Feb 20 '16 at 21:12