Why do we use open intervals in most proofs and definitions? In my class we  usually use intervals and balls in many proofs and definitions, but we almost never use closed intervals (for example, in Stokes Theorem, etc). On the other hand, many books use closed intervals.
Why is this preference? What would happen if we substituted "open" by "closed"?

My guess is that it's because of two related facts.


*

*The advantage of open intervals is that, since every point in the interval has an open neighbourhood within the interval, there are no special points 'at the edge' like in closed intervals, which require being treated differently.

*Lots of definitions rely on the existence of a neighbourhood in their most formal aspect, like differentiability for instance, so key properties within the result may require special formulation at the boundary.
In PDE/functional analysis contexts for example boundaries are very subtle and important objects which are treated separately.
