**Let $f:\mathbb{R}\to \mathbb{R}$ be continuous at $x$ for every $x\in I$ where $I\subset \mathbb R$ could be arbitrary. Does there always exist a function $F:\mathbb{R}\to \mathbb{R}$ differentiable on $I$ and $F'(x) = f(x)$ for every $x \in I$?**

The definition of a primitive is naturally defined on an interval. what sort of weaker result can we obtain under weaker hypotheses?.

If I is an interval or an open set, the answer to the question is positive.

If f is locally Lebesgue integrable,the answer to the question is

also positive.

I have already asked the question here https://math.stackexchange.com/questions/2855483/existence-of-an-antiderivative-function-on-an-arbitrary-subset-of-mathbbr

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