On the robustness of functional equations

In this paper, we study the general question of how characteristics of func tional equations influence whether or not they are robust. We isolate examp les of properties which are necessary for the functional equations to be ro bust. On the other hand, we show other properties which are sufficient for robustness. We then study a general class of functional equations, which ar e of the form For All x, y F[f(x - y), f(x + y); f(x); f(y)] = 0, where F i s an algebraic function. We give conditions on such functional equations th at imply robustness. Our results have applications to the area of self-testing/correcting progra ms. We show that self-testers and self-correctors can be found for many fun ctions satisfying robust functional equations, including algebraic function s of trigonometric functions such as tan x, 1/1+cot x, Ax/1-Ax, cosh x.