The authors reformulate the classical result of D.H. Hyers [Proc. Natl. Acad. Sci. USA 27, 222-224 (1941; Zbl 0061.26403)] concerning the stability of the Cauchy functional equation . Let denotes the Schwartz space of rapidly decreasing functions on and (its dual) the space of tempered distributions. For the pullbacks , , of are defined by
for all test functions . Finally, for , means that for . A distribution is called -additive if
The main result of the paper states that each -additive distribution can be written uniquely in the form with and a bounded measurable function such that . Similar investigations are conducted for a counterpart of the Cauchy equation and its superstability.