I was asked this question by a student I am tutoring and I was left a little puzzled because his textbook only defines antiderivatives on intervals (which leads me to believe its author would answer the question in the title in the affirmative).
To my understanding, finding an antiderivative of $f$ means finding a function $F$ with $F' = f$. It does not matter if the domain of $f%$ is not connected. For example, $\int \frac{dx}{x} $ denotes an antiderivative on all of $\mathbb{R} - 0$, and not just on some arbitrary interval $I \subseteq \mathbb{R} - 0 $. Am I mistaking?