Let $f$ and $g$ be positive, smooth and integrable functions in $\mathbb{R}$, whose derivatives are also integrable. Assume as well that the expression $$ \frac{\sum_{q\in \mathbb{Q}} f(q)}{\sum_{q\in \mathbb{Q}} g(q)} := \lim_{n \to \infty} \frac{\sum_{i =1}^n f(q_i)}{\sum_{i=1}^n g(q_i)} $$ is well defined, where $\{q_i\}_i^\infty = \mathbb{Q}$. I would like to evaluate $$ \left| \frac{\sum_{q\in \mathbb{Q}} f(q)}{\sum_{q\in \mathbb{Q}} g(q)} - \frac{\int f(x) dx}{\int g(x) dx} \right|. $$
When the sum is over a discrete set, I can use the hypothesis to bound the error of Riemann sums by the diameter of the partition generated by the set. What about in such case where the "diameter" is zero?
If $f$ is not identically zero, there is some $a\in\Bbb R$ such that $f(a)>0$. Since $f$ is continuous, there exists $\delta>0$ such that $$ |x-a|<\delta\implies f(x)>\frac{f(a)}{2}>0. $$ Since there are infinitely many rationals $q$ such that $|q-a|<\delta$, we see that $\sum_{q\in\Bbb Q}f(q)=\infty$.