From the author’s summary: In our paper [J. M. Rassias, J. Approx. Th. 57, No. 3, 268-273 (1989; Zbl 0672.41027)] we solved the following Ulam Problem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist” and established results involving a product of powers of norms. In this paper we state and prove a more general version of our above theorem involving a nonnegative real-valued function [S. M. Ulam, “A Collection of Mathematical Problems”, New York (1961; Zbl 0086.241); “Problems in Modern Mathematics”, New York (1964; Zbl 0137.242); “Sets, Numbers and Universes”, Cambridge, MA, (1974; Zbl 0558.00017)].