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.