Is there a function $f \in H(|z|<2)$ such that $$f\left(\frac{1}{n}\right) = \frac{1}{n} \quad \text{ and } \quad f\left(-\frac{1}{n}\right) = \frac{1}{n}$$ for all $n \in \mathbb{N} - \{0\}$? What's an example?
Example of holomorphic function satisfying certain properties
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complex-analysis
holomorphic-functions
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2How could such a function be differentiable at zero? – 2017-01-06
1 Answers
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If $g$ is holomorphic for $\lvert z \rvert < 2$ and $g(1/n) = 1/n$ for all positive integers then $g(z) = z$. To see this note that $g(z) - z$ vanishes on a set that has an accumulation point in the disc. So, to answer your question, a function with the requested property cannot exist.