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Below equation is satisfied.

$$ \int_{0}^{\infty} x^nf(x)dx=0 $$

If $n$ is integer with $n\geq0$, then we can't guarantee $f(x) = 0$ for all positive $x$.

When $n$ is rational number with $n\geq0$ , do we get same result?

If so, what about $n$ is real number with $n\geq 0$?

Above integral is improper Riemann integral.

I forgot one condition. f(x) is continuous function.

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