# 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

## 1 Answer

Firstly, I don't think it entirely makes sense to ask why a property is defined to be local rather than global. Being local or global emerges from the definition itself. Continuity asks about whether a function is well-behaved in a particular way, which is determined locally. We use the word 'continuous' to mean 'well-behaved everywhere', but we also have ways of expressing finer concepts, like 'it's well-behaved here but it isn't there'. So I'd say the main point is that we've chosen the simplest term to refer to the case of good behaviour everywhere, because that is one we want to say a lot.

Secondly, it might make more sense if you switch language. I'm told that some languages (I've forgotten which) have the primary term meaning 'discontinuous' and then make continuity the negation of that. To me that actually seems more natural. The key point of the definition of continuity is that, although we have an intuition for it, it is not straight-forward to define. On the other hand, we have a more concrete idea of what a jump in a function is.

• That's an interesting point! Do you know how they define discontinuity? By negation of the epsilon-delta definition / definition by limit of sequences? Via oscillation? – Stephan Kulla Feb 20 '16 at 13:20
• @tampis I have no direct experience myself, so I do not know. But I assume they define it as we do (ie with the root as the negation of continuity, and other versions shown to be equivalent). – Jessica B Feb 20 '16 at 18:30
• In Russian the term for continuous is "nepreryvny" (непрерывный), which is a negation since the "ne" (не) at the start is "not". It means "no breaks/gaps." But the term discontinuous, for functions, is not gotten by dropping the negative prefix to have "preryvny" (прерывный), but rather is "razryvny" (разрывный). The Russians of course define concepts about continuity exactly as everyone else does; the choice of words has no significance to how the math is handled. – KCd Feb 23 '16 at 7:59