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My question:

Is there a function $f: \mathbb{R} \rightarrow \mathbb{R}$ that nowhere continuous on its domain, but has an antiderivative?

If there is no such a function, is it true to conclude that: to have an antiderivative, $f$ is necessary to be continuous at least at one point on its domain?

Any comments/ inputs are highly appreciated. Thanks in advance.

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    I may be misunderstanding your question, but couldn't you take any nowhere continuous $L^1$ function $f$ and define $F(x) = \int_{-\infty}^x f(t) dt$?2012-02-13
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    The derivative of $F$ is not $f$, I think it's only $f$ a.e.2012-02-13
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    @Chris that $\frac d {dx} F(x) = f(x)$ does not follow from that. I take anti-derivative to mean that there exits a function whose derivative is $f$.2012-02-13
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    I didn't realize that was the definition! Thanks for clearing that up :) I always thought of anti-derivatives as functions $F$ which satisfy $F(b) - F(a) = \int_a^b f(x) dx$, which on reflection really doesn't make much sense.2012-02-13
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    Thanks for your feedbacks! Previously, I also thought the same as Christ, but later on I was in doubt, therefore I asked this question... I think the right answer is "no such function", as verified by Sam and Kahen below, using Baire's theorem2012-02-13

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