Is the Domain of an Antiderivative Always a Subset?

[PFuser1232]

Isn't the domain of the derivative of a function a subset of the domain of the function itself?
Does this mean that the domain of an integrand is always a subset of the corresponding indefinite integral?

 

[Mirero]

We know that if a function $g$ has a derivative $h$, then $dom(h)\subseteq dom(g)$.

Let $f$ be our function and $F$ be any of its corresponding antiderivatives. Then since $F$ has a derivative $f$, it follows that $dom(f)\subseteq dom(F)$.

 

[HallsofIvy]

The domain of the derivative is always a subset of the domain of the function but not necessarily equal. for example, the function f(x)= |x| has domain "all real numbers" while it derivative, f'(x)= 1 if x> 0, -1 if x< 0, as domain "all real numbers except 0".