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exists n f^(n)(x)=0 implies f is a polynomial if f is analytic

Source: VJIMC 2004 2.4

July 12, 2021
algebrapolynomialreal analysis

Problem Statement

Let f:R→Rf:\mathbb R\to\mathbb R be an infinitely differentiable function. Assume that for every x∈Rx\in\mathbb R there is an n∈Nn\in\mathbb N (depending on xx) such that f(n)(x)=0.f^{(n)}(x)=0.Prove that ff is a polynomial.
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