Let , be real normed spaces with complete, and let , be real numbers with . When satisfies the inequality for all , it was shown by T. M. Rassias [Proc. Amer. Math. Soc. 72, 299-300 (1978; Zbl 0398.47040)] that there exists a unique additive mapping such that for all , providing that , where .
The relationship between and was given by the formula . Rassias also proved that if the mapping from to given by is continuous for each fixed , then is linear.
In the present paper the author extends these results to the case , but now the additive mapping is given by , and the corresponding value of is . The author also gives a counterexample to show that the theorem is false for the case , and any choice of when .