Oftentimes we see functions defined on open sets; I never gave this much thought, but now I'd like to get a better understanding.
Why do we define functions over open subsets of $\mathbb{R}^n$?
"Why" is not meant to be a deep philosophical question; what I mean is, for example,
- When is it important to define a function on an open set (and when do we not care whether the set is open or not)?
- What's the motivation for doing so, i.e. what is it that we're going to use about an open set?
- Does using an open set (as opposed to, say, its closure) make anything more convenient or more manageable?
I realize this is quite a general question, and that I didn't give any context; however, I'm hoping that the principles behind defining functions on open sets are "universal" enough that the generality and lack of specific examples won't be too problematic.