As mentioned in this question students sometimes struggle with the fact that continuity is only defined at points of the function's domain. For example the function $f:\mathbb R\setminus\{0\} \to \mathbb R: x \mapsto \tfrac 1x$ is continuous although it has a "jump" at $x=0$ (cf. this answer with more details). So:
Why is continuity only defined on the function's domain? What's the benefit? How should a lecturer answer to such a question of a student?
My attempt to answer the question: I would give two arguments:
- When we take the sequence limit definition of continuity $\lim_{n\to\infty} f(x_n) = f\left(\lim_{n\to\infty} x_n\right) = f(x_0)$, then this definition makes only sense when $x_0 = \lim_{n\to\infty} x_n$ is in the domain of $f$.
- The concept students have in mind is "continuous continuation" and not "continuity". Thus, one have to distinguish between both concepts.
What do think about my answer? Have I missed something or are there other good arguments?
Note: This is another follow up question of How can I motivate the formal definition of continuity? I hope that's okay since I ask here for another aspect of continuity. I want to write an introductory article for continuity. That's the reason why I ask all these questions here...