Principle of permanence

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The function in the middle is given by x2 sin(1/x) for x not equal to 0 and given by 0 for x=0. This cannot be an analytic function, because it is has infinitely many zeros, but is not itself the zero function.

In mathematics, the principle of permanence states that if f(z) is an analytic function defined on an open connected subset U of the complex numbers C, and there exists a convergent sequence {an} having a limit L which is in U, such that f(an) = 0 for all n, then f(z) is uniformly zero on U.

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