A real additive function that is continuous at one point is continuous everywhere and has form . Indeed , and as . Then for every . Additivity gives for every rational number; continuity and the density of the rational numbers extend this identity to every real argument.
Every Lebesgue-measurable additive function is continuous. Small-value preimages are measurable; one of their integer dilates has positive measure, so the Steinhaus theorem puts a neighbourhood of zero in their difference set. Additivity then makes small on that neighbourhood.
If is continuous and is an additive function, then . Additivity proves this first for nonnegative rational numbers; because is continuous, the identity extends to every nonnegative real number.
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Cauchy's functional equation is a well-known functional equation given by: \[ f(x + y) = f(x) + f(y) \] for all real numbers \(x\) and \(y\). This equation describes a function \(f\) that satisfies the property that the value of the function at the sum of two arguments is equal to the sum of the values of the function at each argument.
Nice result on Lebesgue measurable required for uniqueness.