How might one prove the following is either possible or impossible?

Question

$$\int_0^{\infty} g(u)\,du = 0$$ $$\int_0^{\infty} ug(u)\,du = 0$$ where $g(u)$ is continuous over the whole interval and $g(u)$ $\neq 0$ for at least some values of $u$ in the interval.

Does this hold if $g(u)$ diverges as $u \to 0$ but the integral still converges?

Answer 1

This can be taken from any orthogonal polynomial systems with a proper weight, for example, you may check that $$g(u)=\left((\pi-2)u^2-\sqrt\pi u-\frac{\pi-4}{2}\right)e^{-u^2}$$ works

Answer 2

Another option is to use probability distributions. Take $g(u) = f_X(u) - f_Y(u)$ where $f_X \neq f_Y$ have positive support, then

\begin{align} \int_0^\infty g(u) \ du & = \int_0^\infty f_X(u) - f_Y(u) \ du = 0 \\ \int_0^\infty ug(u) \ du & = \int_0^\infty uf_X(u) - uf_Y(u) \ du = E(X) - E(Y). \end{align} Just pick distributions with the same mean.