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Let $f:[0,1]\rightarrow \mathbb{R}$ be a continuous function. Supose that $$\int_{0}^{1}x^{n}f(x)dx=0 $$

$\forall n=0,1,2,...$. Proof that $f(x)=0$.

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closed as off-topic by Brian Fitzpatrick, Davide Giraudo, mookid, Grigory M, Hakim Jun 8 '14 at 10:03

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up vote 6 down vote accepted


These are the steps:

  • Prove that for any polynomial $P$: $$\int_0^1P(x)f(x)dx=0$$
  • Use the Stone–Weierstrass theorem
  • and finally use the fact that if (since $f$ is continuous) $$\int_0^1f^2(x)dx=0$$ then $f$ is the zero function.
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