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.

No such $f(x)$ can exist.  Such a function $f(x)$ would have antiderivative $F(x)$ that satisfies $$F(x_2)-F(x_1)=\frac{1}{x_2-x_1}$$ for all $x_2>x_1>0$.
Take the limit as $x_2\rightarrow \infty$ of both sides; the right side is 0 (hence the limit exists), while the left side is  $$\left(\lim_{x\rightarrow \infty} F(x)\right)-F(x_1)$$ 
Hence $F(x_1)=\lim_{x\rightarrow \infty} F(x)$ for all $x_1$.  Hence $F(x)$ is constant, but no constant function satisfies the relationship desired.
