When discussing properties of functions, we can say "for any smooth function", where smooth means $C^\infty$ or at least "differentiable as many times as we need".
This allows us to present key points that are relevant to our discussion but still acknowledge there are functions that defy it. Examples of using this are "Clairaut's theorem applies to any sufficiently smooth function" (where clarifying what exactly sufficiently smooth means is far more complex that the theorem itself; anyone capable of doing so presumably already knows the theorem).
Likewise for Green and Stokes Theorem: the functions need to be sufficiently smooth. But they also need to be defined without holes: and here we do not seem to have simple terminology. (See below that a function with a single hole at the origin is simply connected but still does not obey these theorems.)
Is there a similar term for the function's domain not having any holes or edges, or at least none that obstruct our point?
I say this because many key theorems in vector calculus apply to smooth functions that are defined everywhere. See When does $\text{div}(\textbf{G})=0$ imply $\textbf{G}=\text{curl}(\textbf{A})$? that saying the domain is simply connected is not sufficient. I'm looking for an expression that, similar to saying "the function is smooth", says the domain is simple enough to avoid obstruction.
Likewise, we need the function to be open. If I want to concisely present these theorems, I'd like a simple term that acknowledges they only apply to functions defined on domains that are sufficiently close to $\mathbb R^n$, without having to enumerate precisely what that means.