The paper discusses the stability of the linear functional equation $ f(x + y) = f(x) + f(y) $. It proves that if a function $ f(x) $ satisfies this equation approximately, then there exists a linear function $ l(x) $ that approximates $ f(x) $. Specifically, for any $ \epsilon > 0 $, there exists a $ \delta > 0 $ such that any $ \delta $-linear transformation $ f(x) $ from a Banach space $ E $ to another Banach space $ E' $ can be approximated by a linear transformation $ l(x) $ with $ \|f(x) - l(x)\| \leq \epsilon $ for all $ x \in E $. The theorem shows that $ \delta $ can be taken equal to $ \epsilon $, and that $ l(x) $ is the unique linear transformation satisfying this inequality. The paper also investigates the continuity of the linear transformation $ l(x) $ when continuity restrictions are placed on $ f(x) $. It proves that if $ f(x) $ is continuous at a single point in $ E $, then $ l(x) $ is continuous everywhere in $ E $. Furthermore, if $ f(tx) $ is continuous in $ t $ for all $ x \in E $, then $ l(x) $ is homogeneous of degree one. The paper concludes by noting that there are many interesting generalizations of the problem, such as considering analogous problems for transformations of topological groups that satisfy the equation for homomorphisms only approximately.The paper discusses the stability of the linear functional equation $ f(x + y) = f(x) + f(y) $. It proves that if a function $ f(x) $ satisfies this equation approximately, then there exists a linear function $ l(x) $ that approximates $ f(x) $. Specifically, for any $ \epsilon > 0 $, there exists a $ \delta > 0 $ such that any $ \delta $-linear transformation $ f(x) $ from a Banach space $ E $ to another Banach space $ E' $ can be approximated by a linear transformation $ l(x) $ with $ \|f(x) - l(x)\| \leq \epsilon $ for all $ x \in E $. The theorem shows that $ \delta $ can be taken equal to $ \epsilon $, and that $ l(x) $ is the unique linear transformation satisfying this inequality. The paper also investigates the continuity of the linear transformation $ l(x) $ when continuity restrictions are placed on $ f(x) $. It proves that if $ f(x) $ is continuous at a single point in $ E $, then $ l(x) $ is continuous everywhere in $ E $. Furthermore, if $ f(tx) $ is continuous in $ t $ for all $ x \in E $, then $ l(x) $ is homogeneous of degree one. The paper concludes by noting that there are many interesting generalizations of the problem, such as considering analogous problems for transformations of topological groups that satisfy the equation for homomorphisms only approximately.