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There is not much (local) information needed to determine an analytic function, as one can see in the following statements. The latter is known as identity theorem.

• \(f^{(k)}(x_0)=g^{(k)}(x_0)\) for one point, then \(f=g\).
• \(f=g\) on a subset with one accumulation point, then \(f=g\).