uniqueness of the fixed point

In section 2.4.3 (Univalent mapping arguement) from 'game theory in SC analysis' by Cachon and Netessine, they claim that if best response function is one-to-one, then there is at most one fixed point(uniqueness of equilibrium). But why? It is possible that we can have x and y where x not equal to y such that x=f(x) and y=f(y). For example, f(x_1,x_2)=(x_2,x_1) is univalent mapping. But when x_1 = x_2=a in R, f(a,a)=(a,a). We can construct a trival game, player x and y write down a number, if x=y, then they both get 10, if not then their payoff is 10-(x-y)^2. In this game, the best response is f(x_1,x_2)=(x_2,x_1) .