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$f$ is a continuous function on $[a,b]$. If $$ \int_a^bx^nf(x)\mathrm{d}x=0,\ n=0,1,2,\cdots. $$ Prove that $f(x)=0$.
A further problem is we only have $$ \int_a^bx^nf(x)\mathrm{d}x=0,\ n=1,2,\cdots. $$ Prove the same result.

I tried to use a polynomial to approach $f(x)$, i.e. $$ \forall \epsilon>0, \exists P(x), \forall x\in[a,b],\ |f(x)-P(x)|\leq\epsilon. $$ Therefore, $$ \int_a^bx^n(P(x)-\epsilon)\mathrm{d}x\leq\int_a^bx^nf(x)\mathrm{d}x\leq \int_a^bx^n(P(x)+\epsilon)\mathrm{d}x. $$ But I do not know how to go on. It seems we only need to see the coefficient of the first term, but I cannot write a clear proof of that. Appreciate any help!

user823011
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    One approach: if you approximate $f(x)$ with a polynomial $f(x) \sim a_0 + a_1x + a_2x^2 + \ldots$ then $0 = \sum \int a_n x^n f(x) \sim \int f(x)^2$. You would need to make this a bit more precise. – Winther Nov 20 '20 at 12:02
  • see also https://math.stackexchange.com/questions/2380609/use-weierstrass-approximation-to-prove-that-fx-0-with-questions-on-the-deta https://math.stackexchange.com/questions/1048287/proof-that-a-continuous-function-is-zero https://math.stackexchange.com/questions/3073455/suppose-f-is-a-lebesgue-integrable-function-on0-1-which-satisfies-int-x?noredirect=1&lq=1 https://math.stackexchange.com/questions/2598875/int-pi-pi-xnfx-0-for-all-non-negative-integer-n-then-f-will-be?noredirect=1&lq=1 https://math.stackexchange.com/questions/1287313/showing-a-function-identically-zero?noredirect=1&lq=1 – Matthew Towers Nov 20 '20 at 12:54

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