I am trying to find the existence of a continuous function defined on $(0,\infty)$ such that \begin{align*} \int_0^\infty x^n f(x)\ dx = \begin{cases} 1, &\mbox{ $n = 0$, }\\ 0, &\mbox { $n \neq 0$, } \end{cases} \end{align*}
where $n \in \mathbf{Z}$.
If restricted only for $n \ge 0$, Stieltjes gave an example of such a function with zero moments. Let \begin{align*} f(x) = \exp\left(-x^{\frac{1}{4}}\right)\sin\left(x^{\frac{1}{4}}\right)\mathbf{1}_{(0,\infty)}(x) \end{align*} and we can show that \begin{align*} \int_0^\infty x^n f(x)\ dx = 0 \end{align*} for all $n \ge 0$. I think I can modify this function slightly such that when $n = 0$ the integral is not zero.
However, if we extend $n$ to take negative values, I see no way out. I am not aware of an existing example. Any idea would be appreciated.