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Let $f:U\to\mathbb{C}$ analytic function, where $U$ is a region. $x_n \to x_0 \in U$ is a real convergent sequence, it is known that $f(x_n)$ is real for all $n$. Is it true $f^{(n)}(x_0)$ is real for all $n$? It is obvious that it is true for $n=0$ and $n=1$, but I could not get further.

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up vote 2 down vote accepted

Let $g(z)=\overline{f(\bar z)}$. It is a holomorphic function and $f(x_n)=g(x_n)$ for all $n$. This however means that $f=g$ (both $f$ and $g$ being holomorphic), hence $f(z)$ is real for real $z$, hence all the derivatives of $f$ are real on the real axis.

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