Stability of the preservation of the equality of distance

We study the stability problem for mappings satisfying the equation ‖f(x−y)‖=‖f(x)−f(y)‖. As a result we obtain the stability of functions preserving the equality of distance, that is, satisfying for all x,y,u,v the condition ‖x−y‖=‖u−v‖⟹‖f(x)−f(y)‖=‖f(u)−f(v)‖.