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Let $f$ be a function. Then $f$ is holomorphic is equivalent to $f$ being analytic which in turn is also equivalent to satisfying the Cauchy-Riemann equations. All three concepts imply infinite differentiability.

My question is what do we need to add to infinite differentiability to recover analyticity/ holomorphicity/Cauchy Riemann equations?

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    Seen [this](http://en.wikipedia.org/wiki/Fabius_function)?2011-08-07
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    ...and [very related.](http://math.stackexchange.com/questions/12989)2011-08-07
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    Thanks a lot, didn't know about these examples!2011-08-07
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    But those are real-analytic, right? So, do you extend them into complex-analytic functions then?2011-08-07
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    I assume you're talking about infinite real differentiability, as being even once complex differentiable entails complex analyticity. I doubt there are any conditions logically weaker than Cauchy-Riemann or equivalents that, when added together with real smoothness, imply holomorphicity, though analysis has a track record of surprising me.2011-08-07
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    I think overall the result of infinite differentiability from analiticity follows from Cauchy's integral formula.2011-08-07
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    Don't you need your functions to be $C^1$ to check holomorphicity with the Cauchy-Riemann equations?2011-08-08

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