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Suppose I have a real-valued function f defined on a complex open connected set D. Am I right in saying that: f is analytic on D if and only if f is n times continuously differentiable on D?

Or does this only apply if f is complex valued?

thanks

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    What do you mean by $n$?2011-04-17
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    Sorry, n is just some natural number > 02011-04-17
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    What do you mean by analytic? What do you mean by differentiable? There is a big difference between considering complex differentiability versus real differentiability; the former is much more rigid than the latter. (I seem to remember recently a thread about this, but I can't find it at the moment)2011-04-17
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    Hmm. Well, I have no control over D; it is either a real open interval or a complex open connected set. I require f(c), ..., f^(n)(c) to exist for certain values of c which are inside D and for the derivatives to be continuous inside D. So instead of saying if D was real that f should be n-times continuously differentiable on D, and saying if D was complex that f should be analytic on D, I thought I'd just say f should be continuously differentiable on D. So I guess by analytic, cts. and differentiable? For differentiabilty, I just need it to exist.. I am not too sure about the differences :|2011-04-17
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    [Complex differentiable](http://mathworld.wolfram.com/ComplexDifferentiable.html) functions are much different from real differentiable functions. Whether $D$ is an **open interval in the real numbers** or it is an **open subset of $\mathbb{C}$** makes a huge difference. On the interval, the only notion of differentiability that makes sense is the real one, and it is **much much weaker** than **analyticity**. In fact even by having *infinitely many derivatives* you cannot guarantee analyticity. On a domain in the complex plane, if you consider it as a domain in $\mathbb{R}^2$, you have the2011-04-17
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    same problem if you use the real-variables definition of differentiability. That is, in the real-variables definition, you can have functions with as many continuous derivatives you like, and still not be analytic. On the other hand, if you consider complex differentiability, just one continuous derivative will tell you that the function is complex analytic. However, as noted below by sos440, it makes relatively little sense to worry about *complex differentiable functions* whose co-domain is purely real.2011-04-17
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    Thanks Willie, that is very helpful.2011-04-17

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