A system of functional equations connected with a problem of aggregation of probability distribution functions is reduced to the functional equation \[ f(x)= {1\over 2} \Biggl\{ f\Biggl({x\over 1- a}\Biggr)+ f\Biggl({x- a\over 1- a}\Biggr)\Biggr\}, \] where \(a\) is a parameter with \(a\in (0, 1)\) and the unknown function \(f: \mathbb{R}\to \mathbb{R}\) is a bounded function with \(f_{|(- \infty, 0)}= 0\) and \(f_{|(1, \infty)}= 1\). The author proves that for any \(a\in (0, 1)\) this functional equation admits exactly one solution. Moreover, this solution is continuous and nondecreasing.