# Necessary conditions for a function to have an antiderivative

Given an interval $$B \subseteq \mathbb{R}$$ and a function $$f : B \to \mathbb{R}$$, what are the necessary conditions (if any) on $$f$$ in order for it to have an antiderivative? Wikipedia lists some examples under the assumption that $$B$$ is an open interval. Are there any others? What if $$B$$ is not open?

• In order to hace derivative at one point, you need to be as near as you want to that point. so $B$ must contain an open interval. Commented Nov 8, 2018 at 19:28
• I did see a related Q on this site, with $B=\Bbb R$. The answers included numerous links to published papers. I got the impression that the answer is a "book" and is still not completed. Commented Nov 8, 2018 at 19:49
• Related, and possibly what @DanielWainfleet was thinking of, is Existence of antiderivative. Commented Nov 8, 2018 at 21:33
• @DaveL.Renfro. Also on this site is the Q of whether a continuous $f:\Bbb R\to \Bbb R$ could be differentiable at each $q\in \Bbb Q$ but not on the irrationals. Regrettably, the Answers addressed $this$ Q instead. Do you anything on the types of subsets of $\Bbb R$ that can ( or can't ) be the set of points of differentiability of a continuous real function? Commented Nov 12, 2018 at 6:03
• @DanielWainfleet: See Characterization of sets of differentiability, where the possible sets of differentiability is discussed rather thoroughly. See also: (a) my answer to Monotone Function, Derivative Limit Bounded, Differentiable - 2; (b) my answer to Construct a function on a bounded interval on $\Bbb{R}$ which is continuous everywhere but differentiable only at irrationals and my comments to Charles Madeline's answer. Commented Nov 12, 2018 at 6:56