Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function
$$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \frac{xy}{(x^2+y^2+1)} \ dx$$
where $x_0$ is an arbitrary but fixed point in $(0,\infty).$
The question is: Can we show that always $g_f \neq f?$
Numerically, it is very easy to set up this problem and it seems to be true that there is no function such that $g_f=f,$ but it is not clear how to conclude this analytically.
Thank you for your time.
