A function/distribution which satisfies an integral equation. (sounds bizzare) I think I need a function, $f(x)$ such that $\large{\int_{x_1}^{x_2}f(x)\,d{x} = \frac{1}{(x_2-x_1)}}$ $\forall x_2>x_1>0$. Wonder such a function been used or studied by someone, or is it just more than insane to ask for such a function? I guess it has to be some distribution, and if so, I'd like to know more about it.

Note that for your function you can easely find $x_1 <x_2 < x_3$ so that
$$\int_{x_1}^{x_2} f(x) \, dx + \int_{x_2}^{x_3} f(x) \, dx \neq \int_{x_1}^{x_3} f(x) \, dx$$
so no such function can exists.
Actually you can prove this way that there exists no measure $m$ so that 
$$ m([x_1,x_2])=\frac{1}{x_2-x_1} \,.$$
so in particular there is no absolutely continuous measure to satisfy that relation.
