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→Properties of analytic functions: fix mistake |
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* Any analytic function is [[smooth function|smooth]].
A polynomial cannot be zero at too many points unless it is the zero polynomial (more precisely, the number of zeros is at most the degree of the polynomial). A similar but weaker statement holds for analytic functions. If the set of zeros of an analytic function ''f'' has an [[accumulation point]] inside its [[___domain (mathematics)|___domain]], then ''f'' is zero everywhere on the [[connected space|connected]] component
More formally this can be stated as follows. If (''r''<sub>''n''</sub>) is a [[sequence]] of distinct numbers such that ''f''(''r''<sub>''n''</sub>) = 0 for all ''n'' and this sequence [[limit of a sequence|converges]] to a point ''r'' in the ___domain of ''D'', then ''f'' is identically zero on the connected component of ''D'' containing
Also, if all the derivatives of an analytic function at a point are zero, then that function is zero. In particular, if an analytic function is zero in the neighborhood of a point, it is zero everywhere.
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