  In this paper we determine the solutions $(\varphi,f_1,f_2)$ of the Pexider
functional equation
\[\varphi\Big(\frac{x+y}2\Big)\big(f_1(x)-f_2(y)\big)=0,\qquad (x,y)\in
I_1\times I_2,\] where $I_1$ and $I_2$ are nonempty open subintervals. Special
cases of the above equation regularly arise in problems with two-variable
means. We show that, under a rather weak regularity condition, the
coordinate-functions of a typical solution of the equation are constant over
several subintervals of their domain. The regularity condition in question will
be that the set of zeros of $\varphi$ is closed. We also discuss particular
solutions where this condition is not met.
