Correct definition of antiderivative function

Question

I am looking for the correct definition of antiderivative function on a set $I\subset \mathbb{R}$ (think of the case where $I = \mathbb{R} / \mathbb{Q}$ )

Let $f: \mathbb{R} \to \mathbb{R}$ and a set $I\subset \mathbb{R}$ (edit $f$ is supposed to be defined on $ \mathbb{R}$ or on an open set containing $I$)

definition 1: an antidérivative function of $f$ on $I$ is a function $F$ differentiable on an open $J$ containing $I$ such that $F '= f$ on $I$

definition 2: an antidérivative function of $f $ on $I$ is a function $F$ (edit $F$ is supposed to be defined on $ \mathbb{R}$ or on an open set containing $I$) differentiable on all points of $I$ such that $F '= f$ on $I$

definition 3: may be

Answer 1

$\newcommand{\R}{\mathbb{R}}$ $%$ Personally, I would say that definition 2 makes more sense; that is:

If we have functions $f,F: \R \rightarrow \R$ and a set $I \subset \R$, then we say $F$ is an antiderivative of $f$ on $I$ if $F$ is differentiable on $I$ and $F'(x) = f(x)$ for every $x \in I$.

I don't see a reason why we should require $F$ to be differentiable on open sets around $I$. However, this requirement may be useful for applications of this definition, of which I know none. As I pointed out in a comment, it really depends on how one plans to use this definition.