Consider the integral
$$I = \int_{-\infty}^{\infty} f(x)g(x) \mathrm{d}x.$$
Suppose that
1) $I$ is well defined;
2) $f$ is continuos on $(-\infty, \infty)$;
3) $g$ is not well-defined for infinitely many values on $(-\infty, \infty)$.
My question is: can a dyadic decomposition work for estimating $I$ ?