The distribution $g(x)$ has the following properties:


$$\int\_{-\infty}^{\infty}g'(x)f(x)dx=f(\pi g(0)-1/2)-f(\pi g(0)+1/2)$$


for any analytic $f(x)$.


How can I find $g(x)$?



Are there solutions for this functional equation?
If your equation extends to smooth test functions, this is just $g'=\delta\_{\pi g(0)-1/2}-\delta\_{\pi g(0)+1/2}$, which is equivalent to $g=c+\mathbb1\_{[\pi g(0)-1/2\ ,\ \pi g(0)-1/2]}$ for some $c$ . If you chose $c$ such that $|c|>\frac1{2\pi}$, then $g=c+\mathbb1\_{[c\pi-1/2\ ,\ c\pi-1/2]}$ is a solution.


