Consider a random variable $X$ distributed as $H(\cdot)$ on $[\underline{x}, \overline{x}]$. Suppose $H(\cdot)$ also has a "well defined" density $h(\cdot)$ that is strictly bounded away from zero on the entire support. In other words. $H(\cdot)=\Pr(X\leq \cdot)$ is a strictly increasing function.
Let $\theta(a,b)=\frac{H(a)}{H(b)}$ for any $a\in[\underline{x}, \overline{x}]$ and $b\in[\underline{x}, \overline{x}]\in\mathbb{R}_{+}$. Suppose we know the function $\theta(a,b)$ but neither $H(\cdot)$ nor $[\underline{x}, \overline{x}]$. How can we determine $H(\cdot)$ from the ratio?
One way to recover $H(\cdot)$ from the ratio is to consider the limit $\theta(a,\overline{x}):=\lim_{b\rightarrow \overline{x}}\theta(a,b)$. Then $\theta(a,\overline{x})=\frac{H(a)}{H(\overline{x})}=H(a)$.
But I am interested in finding a solution that does not rely on such "limit" argument. Thanks.