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STATEMENT: Let $f(x)$ be a continuous function on $\mathbb{R}$ such that for any polynomial $P(x)$ we have

$$\int_\mathbb{R} f(x)P(x) \, dx=0$$

Show that $f(x)$ is identically zero.

QUESTION: This is an old prelim problem and I'm having trouble showing it. The usual problem restricts to a bounded domain $[a,b]$ from which we can apply stone weirstrass and some manipulation to get the desired result, but reducing to that case doesn't seem possible here, so does anyone have a suggestion on how to proceed? Some hints would be appreciated.

EDIT: Note that this is not a duplicate question. Does $\int_{\mathbb R} f(x)x^n dx = 0$ for $n=0,1,2,\ldots$ imply $f=0$ a.e.?. The other post does not assume that $f$ is continuous.

Michael Hardy
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Enigma
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    Does this provide a counterexample (extended by symmetry)? And this? – Dabouliplop Aug 09 '17 at 01:47
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    Just to clarify, you interpret $\int_\mathbb{R}$ as $\lim_{N \to \infty} \int_{-N}^N$ and not the Lebesgue integral, correct? – mathworker21 Aug 09 '17 at 03:01
  • @mathworker21 wouldn't the two definitions coincide since our functions are continuous in this case? But in any case I assumed it was the Lebesgue integral; there was no mention of anything else for the problem. – Enigma Aug 09 '17 at 03:39
  • @Ideophage Thanks that works. Guess they didn't really check the problems carefully. – Enigma Aug 09 '17 at 03:40

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